IRLF 


*B    ML 


'.OWN  &  SHARPK  MFC.  CO 
R0V1D]     (  ;•:,  R.  I.,  U.  S.  A. 


Mechanics  Department 


-  _  ;  _  • 

/  f 

'-PRACTICAL  TREATISE 


\    *  ••  *  t  *>  »*  ^  *•  • 

",•*•*••     •*  *•*  ' 


SEVENTH  EDITION. 


BROWN  &  SHARPE  MANUFACTURING  CO. 

PROVIDENCE,  R.  I.,  U.  S.  A. 
1902 


Engineering 
Library 


MECHANICS   DEPT/ 

COPYRIGHT, 

1886,  1887,  1892,  1893,  1896,  1900,  1902, 

BY 

BROWN  &  SHARPE  MFG.  CO. 

Registered  at  Stationers'  Hall,  London,  Eng. 

All  rights  reserved. 


PREFACE. 


This  Book  is  made  for  men  in  practical  life ;  for  those  that 
would  like  to  know  how  to  construct  gear  wheels,  but  whose 
duties  do  not  afford  them  sufficient  leisure  to  acquire  a  technical 
knowledge  of  the  subject. 


CONTENTS. 


PART     I . 
CHAPTER  I. 

PAGE. 

Pitch  Circle — Pitch — Tooth — Space — Addendum  or  Face — 

Flank — Clearance 1 

CHAPTER  II. 

Classification — Sizing  Blanks  and  Tooth  Parts  from  Linear 

or  Circular  Pitch — Center  Distance 5 

CHAPTER  III. 
Single  Curve  Gears  of  30  Teeth  and  more ,       9 

CHAPTER  IV. 

Rack  to  Mesh  with  Single  Curve  Gears  having  30  Teeth  and 

more 12 

CHAPTER  V. 

Diametral  Pitch — Sizing  Blanks  and  Teeth — Distance  be- 
tween the  Centers  of  Wheels 16 

CHAPTER  VI. 

Single-Curve  Gears,  having  Less  than  30  Teeth — Gears  and 

Racks  to  Mesh  with  Gears  having  Less  than  30  Teeth...     20 

CHAPTER  VII. 
Double-Curve  Teeth— Gear  of  15  Teeth— Rack 25 

CHAPTER  VIII. 

Double-Curve  Gears,  having  More  and  Less  than  15  Teeth 

— Annular  Gears ,  30 


VI  CONTENTS. 

CHAPTER  IX. 

PAGE. 

Bevel  Gear  Blanks 34 

CHAPTER  X. 
Bevel  Gears — Form  and  Size  of  Teeth — Cutting  Teeth 41 

CHAPTER  XI. 
Worm  Wheels — Sizing  Blanks  of  32  Teeth  and  more G3 

CHAPTER  XII. 

Sizing  Gears  when  the  Distance  between  the  Centers  and  the 
Ratio  of  Speeds  are  fixed — General  Remarks — Width  of 
Face  of  Spur  Gears — Speed  of  Gear  Cutters — Table  of 
Tooth  Parts 79 


PART     II. 

CHAPTER  I. 
Tangent  of  Arc  and  Angle 8? 

CHAPTER  II. 

Sine,  Cosine  and   Secant — Some   of   their   Applications  in 

Machine  Construction 93 

CHAPTER  III. 

Application  of  Circular  Functions — Whole  Diameter  of  Bevel 

Gear  Blanks— Angles  of  Bevel  Gear  Blanks 100 

CHAPTER  IV. 
Spiral  Gears — Calculations  for  Pitch  of  Spirals 107 

CHAPTER  V. 

Examples  in  Calculations  of  Pitch  of  Spirals — Angle  of 
Spiral — Circumference  of  Spiral  Gears — A  few  Hints 
on  Cutting Ill 


CONTENTS.  VII 

CHAPTER  VI. 

Normal  Pitch  of  Spiral  Gears — Curvature  of  Pitch  Surface 

— Formation  of  Cutters 114 

CHAPTER  VII. 
Cutting  Spiral  Gears  in  a  Universal  Milling  Machine..., 120 

CHAPTER  VIII. 
Screw  Gears  and  Spiral  Gears — General  Remarks.,. 12? 

CHAPTER  IX. 

Continued  Fractions — Some  Applications  in  Machine  Con- 
struction   130 

CHAPTER  X. 
Angle  of  Pressure  135 

CHAPTER  XI. 
In ternal  Gears — Tables — Index 137 

CHAPTER  XII. 
Strength  of  Gears— Tables  .of  Tooth  Parts , 140 


PART   I. 


CHAPTER/I..   •:';•  !•' 

PITCH  CIRCLE,  PITCH,  TOOTS;  SP.ACE,:.4I(BEJin|E-.e8/F.ACE,  FLAHK, 

'CLEARANCE. 


Let  two  cylinders,  Fief.  1,  touch  each  other,  their  Original 
axes  be  parallel  and  the  cylinders  be  on  shafts,  turning 
freely.  If,  now,  we  turn  one  cylinder,  the  adhesion  of 
its  surface  to  the  surface  of  the  other  cylinder  will 
make  that  turn  also.  The  surfaces  touching  each 
other,  without  slipping  one  upon  the  other,  will  evi- 
dently move  through  the  same  distance  in  a  given 

A.-  rm  •  £  -i   -          11     i   7.  Linear  Veloci 

time.     1ms  suriace  speed  is  called  linear  velocity.         ty. 

TANGENT    CYLINDERS. 


Fig.l 

LINEAE  VELOCITY  is  the  distance  a  point  moves  along 
a  line  in  a  unit  of  time. 

The  line  described  by  a  point  in  the  circumference 
of  either  of  these  cylinders,  as  it  rotates,  may  be  called 
an  arc.  The  length  of  the  arc  (which  may  be  greater 
or  less  than  the  circumference  of  cylinder),  described 
in  a  unit  of  time,  is  the  velocity.  The  length,  expressed 
in  linear  units,  as  inches,  feet,  etc.,  is  the  linear  velocity. 


BROWN    &    SHARPE    MFG.    CO. 


Relative  An 
gular  Velocity 


The  length,  expressed  in  angular  units,  as  degrees,  is 
the  angular  velocity. 

If  now,  instead  of  1°  we  take  360°,  or  one  turn,  as 
Ve  tbe  angular  unit,  and  1  minute  as  the  time  unit,  the 
angular  velocity^ will  bs  expressed  in  turns  or  revolu- 
tions per  minute. : 

/If  t^s^Jttfo/kyUnders  are  of  the  same  size,  one  will 
make  the  same  number  of  turns  in  a  minute  that  the 
other  makes.  If  one  cylinder  is  twice  as  large  as  the 
other,  the  smaller  will  make  two  turns  while  the  larger 
makes  one,  but  the  linear  velocity  of  the  surface  of 
each  cylinder  remains  the  same. 

This  combination  would  be  very  useful  in  mechan- 
ism if  we  could  be  sure  that  one  cylinder  would  always 
turn  the  other  without  slipping. 


O/HCLE 


ig.  3 


In  the  periphery  of  these  two  cylinders,  as  in  Fig. 
2,  cut  equidistant  grooves.     In  any  grooved  piece  the 
Land.  places  between  grooves  are  called  lands.     Upon  the 

Addendum,    lands  add  parts  ;   these  parts  are  called  addenda.     A 
Tooth.  land  and  its  addendum  is  called  a  tooth.     A  toothed 

Gear.  cylinder  is  called  a  gear.     Two  or  more   gears  with 

Train.  teeth  interlocking  are  called  a  train.     A  line,  c  c',  Fig. 


PROVIDENCE,    R.    I. 


3 


Addendum 

Circle. 


2  or  3,  between  the  centers  of  two  wheels  is  called  the    Line  of  Cen. 
line  of  centers.     A  circle  just  touching  the  addenda ters' 
is  called  the  addendum  circle. 

The  circumference  of  the  cylinders  without  teeth  is 
called   the  pitch  circle.     This  circle  exists  geometri- pitch  circle- 
cally  in  every  gear  and  is  still  called  the  pitch  circle    pitch    circle 
or  the  primitive  circle.    In  the  study  of  gear  wheels,  it  j^e  ^Primitive 
is  the  problem  so  to  shape  the  teeth  that  the   pitch  circle, 
circles  will  just  touch  each  other  without  slipping. 

On  two  fixed  centers  there  can  turn  only  two  circles, 
one  circle  on  each  center,  in  a  given  relative  angular 
velocity  and  touch  each  other  without  slipping. 


4  BROWN    &    SHARPE    MFG.    CO. 

Space.  The  groove  between  two  teeth  is  called  a  space.    In 

cut  gears  the  width  of  space  at  pitch  line  and  thickness 

of  tooth  at  pitch  line  are  equal.     The  distance  between 

the  center  of  one  tooth  and  the  center  of  the  next  tooth, 

Linear  or  cir- measured  along:  the  pitch  line,  is  the  linear  or  circular 

cular  Pitch.  . 

pitch;  that  is,  the  linear  or  circular  pitch  is  equal  to  a 
Tooth  Thick- tooth  and  a  space;  hence,  the  thickness  of  a  tooth  at 
the  pitch  line  is  equal  to  one-half  the  linear  or  circular 
pitch. 

tionsbofeparts     Let  D  =  diameter  of  addendum  circle. 
G°erareeth  and       "   D'=  diameter  of  pitch  circle. 
"   P'— linear  or  circular  pitch. 
"   t  =  thickness  of  tooth  at  pitch  line. 
"   s  =  addendum  or  face,  also  length   of  working 

part  of  tooth  below  pitch  line  or  flank. 
"   2s  =  D"  or  twice  the  addendum,  equals  the  work- 
ing depth  of  teeth  of  two  gears  in  mesh. 
"  /=  clearance  or  extra  depth  of  space  below  work- 
ing depth. 

"  s-f/— depth  of  space  below  pitch  line. 
"   D"+/'=  whole  depth  of  space. 
"   N  =  number  of  teeth  in  one  gear. 
"   7r=3.1416  or  the  circumference  when  diameter 

isl. 

Pf  is  read  "P  prime."     D"  is  read  "D  second."     it  is 
read  "pi." 

TO  find  the     If  we  multiply  the  diameter  of  any  circle  by  n,  the 
andUDiameterprodnct  will  be  the  circumference  of  this  circle.     If  wo 
divide  the  circumference  of  any  circle  by  ft,  the  quo- 
tient will  be  the  diameter  of  this  circle. 

Pitch  Point.  The  pitch  point  of  the  side  of  a  tooth  is  the  point  at 
which  the  pitch  circle  or  line  meets  the  side  of  the 
»tooth.  A  gear  tooth  has  two  pitch  points. 


CHAPTER  II. 

CLASSIFICATION-SIZING  BLANKS  AND  TOOTH  PARTS  FROM 
CIRCULAR  PITCH— CENTRE  DISTANCE— PATTERN  GEARS. 


If  we  conceive  the  pitch  of  a  pair  of  gears  to  be  ^^J^f  of 
made  the  smallest  possible,  we  ultimately  come  to  the 
conception  of  teeth  that  are  merely  lines  upon  the 
original  pitch  surfaces.  These  lines  are  called  ele- 
ments of  the  teeth.  Gears  may  be  classified  with 
reference  to  the  elements  of  their  teeth,  and  also  with 
reference  to  the  relative  position  of  their  axes  or  -shafts. 
In  most  gears  the  elements  of  teeth  are  either  straight 
lines  or  helices  (screw-like  lines). 

PART  I.  of  this  book,  treats  upon  THREE  KINDS  OF 

GEARS. 

First — SPUR  GEARS  ;  those  connecting  parallel  shafts  sPur  Gears- 
and  whose  tooth  elements  are  straight. 

Second — BEVEL  GEARS;  those  connecting  shafts Bevel  Geara- 
whose  axes  meet  when  sufficiently  prolonged,  and  the 
elements  of  whose  teeth  are  straight  lines.  In  bevel 
gears  the  surfaces  that  touch  each  other,  without 
slipping,  are  upon  cones  or  parts  of  cones  whose 
apexes  are  at  the  same  point  where  axes  of  shafts  meet. 

Third — SCREW  OR  WORM  GEARS;  those  connecting 
shafts  that  are  not  parallel  and  do  not  meet,  and  the 
elements  of  whose  teeth  are  helical  or  screw-like. 

The  circular  pitch  and  number  of  teeth  in  a  wheel 
being  given,  the   diameter   of  the  wheel  and  size  of      Blanks,  &o. 
tooth  parts  are  found  as  follows : 

Dividing  by  3.1416  is  the  same  as  multiplying  by 

3T4T6-  Now  3.1416— -3183;  nence>  multiply  the  cir- 
cumference of  a  circle  by  .3183  and  the  product  will  be 
the  diameter  of  the  circle.  Multiply  the  circular  pitch 
by  .3183  and  the  product  will  be  the  same  part  of  the 


6  BROWN    &    SHARPE    MFG.    CO. 

diameter  of  pitch  circle  that  the  circular  pitch  is  of  the 
A  Diameter  circumference  of  pitch  circle.     This  part  is  called  the 
pitch,  or  Mod  module  of  the  pitch.     There  are  as  many  modules  con- 
tained in  the  diameter  of  a  pitch  circle  as  there  are 
teeth  in  the  wheel. 

Most  mechanics  make  the  addendum  of  teeth  equal 
the  module.  Hence  we  can  designate  the  module  by 
the  same  letter  as  we  do  the  addendum;  that  is,  let  s  = 
the  module. 

.3183  P'=s,  or  circular  pitch  multiplied  by  .3183 =s, 
or  the  module. 

Diameter  of     Ns  — Df,  or  number  of  teeth  in  a  wheel,  multiplied 
by  the  module,  equals  diameter  of  pitch  circle. 

(N  +  2)  5  =  D,  or  add  2  to  the  number  of  teeth,  mul- 
et^hole  Diam"  tiply  the  sum  by  the  module  and  product  will  be  the 
whole  diameter. 

T$=f,  or  one  tenth  of  thickness  of  tooth  at  pitch  line 
clearance.       equals  amount  added  to  bottom  of  space  for  clearance. 
Some  mechanics  prefer  to  make  f  equal  to  -^  of  the 
working  depth  of  teeth,  or  .0625  D".     One-tenth  of  the 
thickness  of  tooth  at  pitch-line  is  more  than  one-six- 
teenth of  working  depth,  being  .07854  D". 

Example.  Example.— Wheel  30  teeth,  1£"  circular  pitch.    P'= 

sizesof  Blank1-5"?  then  *=-75"  or  thickness  of  tooth  equals  f".     s  = 
panrtds  foVaelr  1-5'/ x  -3183  —4?75  =  module  for  iy\  P'.     (See  table  of 

to^ftSmliS*00*11  Parts>  PaSes  144-U7- 

Pitch.  D'=30x.4775w=14.325f'==diameter  of  pitch-circle. 

D  =  (30+2)  x  .4775"=  15.280"=:diameter  of  adden- 
dum circle,  or  the  diameter  of  the  blank. 

/=TV  of  .75ff=.075"=  clearance  at  bottom  of  space. 

D"=2x.4775"=.9549"=  working  depth  of  teeth. 

D"+/=2x. 4775"+. 075"=  1. 0299"—  whole  depth  of 
space. 

s+/=.4775"+. 075 "=.5525"=: depth  of  space  inside 
of  pitch-line. 

D"=2s  or  the  working  depth  of  teeth  is  equal  to  two 
modules. 

In  making  calculations  it  is  well  to  retain  the  fourth 
place  in  the  decimals,  but  when  drawings  are  passed 
into  the  workshop,  three  places  of  decimals  are  suffi- 
cient. 


PROVIDENCE,    R.    I. 


FIG.  5,  SPUR  GEARING. 


8  BROWN   &    SHARPE   MFC.    CO. 


Distance  be-      ^he  distance  between  the  centers  of  two  wheels  is 

IW6611    c©HLt~rs 

of  two  Gears,  evidently  equal  to  the  radius  of  pitch-circle  of  one  wheel 
added  to  that  of  the  other.  The  radius  of  pitch-circle 
is  equal  to  s  multiplied  by  one-half  the  number  of  teeth 
in  the  wheel. 

Hence,  if  we  know  the  number  of  teeth  in  two  wheels, 
in  mesh,  and  the  circular  pitch,  to  obtain  the  distance 
between  centers  we  first  find  s  ;  then  multiply  s  by  one- 
half  the  sum  of  number  of  teeth  in  both  wheels  and  the 
product  will  be  distance  between  centers. 

Example  —  What  is  the  distance  between  the  centers 
of  two  wheels  35  and  60  teeth,  1J"  circular  pitch.  "We 
first  find  s  to  be  lj"x  .3183=.3979".  Multiplying  by 
47.5  (one-half  the  sum  of  35  and  60  teeth)  we  obtain 
18.899"  as  the  distance  between  centers. 

ShHnTag  e  f°n  Pattern  Gears  should  be  made  large  enough  to 
Gear  Castings,  allow  for  shrinkage  in  casting.  In  cast-iron  the  shrinkage 
is  about  ^  inch  in  one  foot.  For  gears  one  to  two  feet 
in  diameter  it  is  well  enough  to  add  simply  -^  of 
diameter  of  finished  gear  to  the  pattern.  In  gears 
about  six  inches  diameter  or  less,  the  moulder  will 
generally  rap  the  pattern  in  the  sand  enough  to  make 
any  allowance  for  shrinkage  unnecessary.  In  pattern 
gears  the  spaces  between  teeth  should  be  cut  wider 
than  finished  gear  spaces  to  allow  for  rapping  and  to 
avoid  having  too  much  cleaning  to  do  in  order  to  have 
gears  run  freely.  In  cut  patterns  of  iron  it  is  generally 
Metal  Pattern  enough  to  make  spaces  .015"  to  .02"  wider.  This 

Gr6£tTS 

makes  clearance  .03"  to  .04"  in  the  patterns.  Some 
moulders  might  want  .06"  to  .07"  clearance. 

Metal  patterns  should  be  cut  straight  ;  they  work 
better  with  no  draft.  It  is  well  to  leave  about  .005"  to 
be  finished  from  side  of  patterns  after  teeth  are  cut  ; 
this  extra  stock  to  be  taken  away  from  side  where 
cutter  comes  through  so  as  to  take  out  places  where 
stock  is  broken  out.  The  finishing  should  be  done 
with  file  or  emery  wheel,  as  turning  in  a  lathe  is  likely 
to  break  out  stock  as  badly  as  a  cutter  might  do. 

If  cutters  are  kept  sharp  and  care  is  taken  when 
coming  through  the  allowance  for  finishing  is  not  nec- 
essary and  the  blanks  may  be  finished  before  they  are 
cut. 


CHAPTER   III. 
SINGLE-CURVE  GEARS  OF  30  TEETH  AND  MORE. 


Single-curve  teeth  are  so  called  because  they  have  Tf^le  Curve 
but  one  curve  by  theory,  this  curve  forming  both  face 
and  flank  of  tooth  sides.  In  any  gear  of  thirty  teeth 
and  more,  this  curve  can  be  a  single  arc  of  a  circle 
whose  radius  is  one-fourth  the  radius  of  the  pitch 
circle.  In  gears  of  thirty  teeth  and  more,  a  fillet  is 
added  at  bottom  of  tooth,  to  make  it  stronger,  equal 
in  radius  to  one-seventh  the  widest  part  of  tooth  space. 

A  cutter  formed  to  leave  this  fillet  has  the  advantage 
of  wearing  longer  than  it  would  if  brought  up  to  a 
corner. 

In  gears  less  than  thirty  teeth  this  fillet  is  made  the 
same  as  just  given,  and  sides  of  teeth  are  formed  with 
more  than  one  arc,  as  will  be  shown  in  Chapter  YI. 

Having  calculated  the  data  of  a  gear  of  30  teeth,  J    Example  of  a 
inch  circular  pitch  (as  we  did  in  Chapter  II.  for  1£"  =%"• 
pitch),  we  proceed  as  follows  : 

1.  Draw  pitch  circle  and  point  it  off  into  parts  equal      Geometrical 

T     ..„  .,          .         T  . .    1  Construction. 

to  one-halt  the  circular  pitch.  Fig.  6. 

2.  From  one*  of  these  points,  as  at  B,  Fig.  6,  draw 
radius  to  pitch  circle,  and  upon  this  radius  describe  a 
semicircle ;  the  diameter  of  this  semicircle  being  equal 
to  radius  of  pitch  circle.      Draw  addendum,  working 
depth  and  whole  depth  circles. 

3.  From  the  point  B,  Fig.  6,  where  semicircle,  pitch 
circle  and  outer  end  of  radius  to  pitch  circle  meet,  lay 
off  a  distance  upon  semicircle  equal  to  one-fourth  the 
radius  of  pitch  circle,  shown  in  the  figure  at  BA,  and 
is  laid  off  as  a  chord. 

4.  Through  this  new  point  at  A,  upon  the  semicircle, 
draw  a  circle  concentric  to  pitch  circle.     This  last  is 


10 


BROWN    &    SHARPE    MFG.    CO. 


GEAR,  SO  TEETH, 
"  CIRCULAR  PITCH 

P'=     "  or  .75" 


t=    .375" 

S=   .2387" 

D"=    .4775" 

8+/=    .2762" 

D"+/=    .5150" 

D'=  7.1610" 

D  =  7.63S4" 


SINGLE   CURVE    GEAR. 


PROVIDENCE,    R.    I.  11 

called  the  base  circle,  and  is  the  one  for  centers  of 
tooth  arcs.  In  the  'system  of  single  curve  gears  we 
have  adopted,  the  diameter  of  this  circle  is  .968  of  the 
diameter  of  pitch  circle.  Thus  the  base  circle  of  any 
gear  1  inch  pitch  diameter  by  this  system  is  .968". 
If  the  pitch  circle  is  2"  the  base  circle  will  be  1.936." 

5.  With  dividers  set  to  one-quarter  of  the  radius  of 
pitch  circle,  draw  arcs  forming  sides  of  teeth,  placing 
one  leg  of  the  dividers  in  the  base  circle  and  letting 
the  other  leg  describe  an  arc  through  a  point  in  the 
pitch  circle  that  was  made  in  laying  off  the  parts  equal 
to  one-half  the  circular  pitch.     Thus  an  arc  is  drawn 
about  A  as  center  through  B. 

6.  With  dividers  set  to  one-seventh  of  the  widest  part 
of  tooth  space,  draw  the  fillets  for  strengthening  teeth 
at  their  roots.     These  fillet  arcs  should  just  touch  the 
whole   depth   circle   and   the   sides   of   teeth   already 
described. 

Single  curve  or  involute  gears  are  the   only  gears  Invoiutea3ea?- 
that  can  run  at  varying  distance  of  axes  and  transmit mg- 
unvarying  angular  velocity.     This  peculiarity  makes 
involute  gears   specially  valuable  for  driving  rolls  or 
any  rotating   pieces,   the    distance   of  whose    axes   is 
likely  to  be  changed. 

The  assertion  that  gears  crowd  harder  on  bearings 
when  of  involute  than  when  of  other  forms  of  teeth, 
has  not  been  proved  in  actual  practice. 

Before  taking  next  chapter,  the  learner  should  make    Practice,  be- 
several  drawings   of  gears  30  teeth  and  more.     Say  next  chapter. 
make  35  and  70  teeth  \\"  P'.     Then  make  40  and  65 
teeth  |"  P'. 

An  excellent  practice  will  be  to  make  drawing  on 
cardboard  or  Bristol-board  and  cut  teeth  to  lines,  thus 
making  paper  gears ;  or,  what  is  still  better,  make  them 
of  sheet  metal.  By  placing  thesa  in  mesh  the  learner 
can  test  the  accuracy  of  his  work. 


12 


CHAPTER    IV. 

RACK  TO  MESH  WITH  SINGLE-CURVE  GEARS  HAVING 
30  TEETH  AND  MORE. 


madeaprepam-     Tllis  &ear  (Fi£-  7) is  made  precisely  the  same  as  gear 
a°RackdrawinS  *n  Chapter  III.    It  makes  no  difference  in  which  direc- 
tion the  construction  radius  is  drawn,  so  far  as  obtain- 
ing form  of  teeth  and  making  gear  are  concerned. 

Here  the  radius  is  drawn  perpendicular  to  pitch  line 
of  rack  and  through  one  of  the  tooth  sides,  B.  A  semi- 
circle is  drawn  on  each  side  of  the  radius  of  the  pitch 
circle. 

The  points  A  and  A'  are  each  distant  from  the  point 
B,  equal  to  one-fourth  the  radius  of  pitch  circle  and 
correspond  to  the  point  A  in  Fig.  6. 

In  Fig.  7  add  two  lines,  one  passing  through  B  and 
A  and  one  through  B  and  A;.  These  two  lines  form 
angles  of  75^°  (degrees)  with  radius  BO.  Lines  BA 
and  BA'  are  called  lines  of  pressure.  The  sides  of 
rack  teeth  are  made  perpendicular  to  these  lines. 
Rack.  A  Rack  is  a  straight  piece,  having  teeth  to  mesh 

with  a  gear.  A  rack  may  be  considered  as  a  gear  of 
infinitely  long  radius.  The  circumference  of  a  circle 
approaches  a  straight  line  as  the  radius  increases,  and 
when  the  radius  is  infinitely  long  any  finite  part  of  the 
construction  circumference  is  a  straight  line.  The  pitch  line  of  a 

of  Pitch  Line  of 

Rack.  rack,  then,  is  merely  a  straight  line  just  touching  the 

pitch  circle  of  a  gear  meshing  with  the  rack.      The 

thickness    of    teeth,    addendum    and  depth   of    teeth 

below  pitch  line  are  calculated  the  same  as  for  a  wheel. 

(For  pitches  in  common  use,  see  table  of  tooth  parts.) 

The  term  circular  pitch  when  applied  to  racks  can  be 

more  accurately  replaced   by   the   term   linear  pitch. 

Linear  applies  strictly  to  a  line  in  general  while  circular 

pertains  to  a  circle.     Linear  pitch  means  the  distance 

between  tbe  centres   of  two   teeth  on  the  pitch  line 

whether  the  line  is  straight  or  curved. 


PROVIDENCE,    E.    I. 


13 


A  rack  to  mesh  with  a  single-curve  gear  of  30  teeth 
or  more  is  drawn  as  follows : 

1.  Draw  straight  pitch  line  of  rack ;  also  draw  ad- 
dendum line,  working  depth  line  and  whole  depth  line, 
each  parallel  to  the  pitch  line  (see  Fig.  7). 


Back. 
Fig.  7. 


RACK  %   CIRCULAR  FITCH. 


RACK  TO  MESH  WITH   SINGLE  CURVE  GEAR 
HAVING  30  TEETH  AND  MORE. 


14  BROWN    &    SHARPE    MFG.    CO. 

2.  Point  off  the  pitch  line  into  parts  equal  to  one- 
half  the  circular  pitch,  or  =  t. 

3.  Through  these  points  draw  lines  at  an  angle  of 
7o£°  with  pitch  lines,  alternate  lines  slanting  in  oppo- 
site directions.     The  left-hand  side  of  each  rack  tooth 
is  perpendicular  to  the  line  BA.     The  right-hand  side 
of  each  rack  tooth  is  perpendicular  to  the  line  BA'. 

4.  Add  fillets  at  bottom  of  teeth  equal  to  ^  of  the 
width  of  spaces  between  the  rack  teeth  at  the  adden- 
dum line. 

sid^of6  nick  The  sketch,  Fig.  8,  will  show  how  to  obtain  angle  of 
sides  of  rack  teeth,  directly  from  pitch  line  of  rack, 
without  drawing  a  gear  in  mesh  with  the  rack. 


Upon  the  pitch  line  b  b',  draw  any  semicircle— 
b  a  a'  b'.  From  point  b  lay  off  upon  the  semicircle 
the  distance  b  a,  equal  to  one-quarter  of  the  diameter 
of  semicircle,  and  draw  a  straight  line  through  b  and  a. 

This  line,  b  a,  makes  an  angle  of  75  J°  with  pitch  line 
b  5',  and  can  be  one  side  of  rack  tooth.  The  same 
construction,  b'  a',  will  give  the  inclination  75£°  in  the 
opposite  direction  for  the  other  side  of  tooth. 

The  sketch,  Fig.  9,  gives  the  angle  of  sides  of  a  tool 
for  planing  out  spaces  between  rack  teeth.  Upon  any 
line  OB  draw  circle  OABA'.  From  B  lay  off  distance 
BA  and  BA',  each  equal  to  one-quarter  of  diameter  of 
the  circle. 

Draw  lines  OA  and  OA'.  These  two  lines  form  an 
angle  of  29°,  and  are  right  for  inclination  of  sides  of 
rack  tool. 


PROVIDENCE,    R.    I. 

Make  end  of  rack  tool  .31  of  circular  pitch,  and  then 
round  the  corners  of  the  tool  to  leave  fillets  at  the 
bottom  of  rack  teeth. 

Thus,  if  the  circular  pitch  of  a  rack  is  1J"  and  we 
multiply  by  .31,  the  product  .465"  will  be  the  width  of 
tool  at  end  for  rack  of  this  pitch  before  corners  are 
taken  off.  This  width  is  shown  at  x  y. 


15 


A  Worm  is  a  screw  that  meshes  with  the  teeth  of  a 
gear. 

This  sketch  and  the  foregoing  rule  are  also  right  for  Worm  Thread 
a    worm-thread  tool,  but    a  worm-thread  tool  is  not 
usually  rounded  for  fillet.     In  cutting  worms,  leave 
width  of   top   of   thread  .335    of   the   circular   pitch. 
When  this  is  done,  the  depth  of  thread  will  be  right. 


SKETCH    OF    WORM    THREAD 


16 


CHAPTER  V. 

DIAMETRAL  PITCH— SIZING  BLANKS  AND  THE  TEETH  OF  SPUR  GEARS 
—DISTANCE  BETWEEN  THE  CENTRES  OF  WHEELS. 


necessary  to      ^n  making  drawings  of  gears,  and  in  cutting  racks, 


^  *s  necessary  to  know  the  circular  pitch,  both  on 
account  of  spacing  teeth  and  calculating  their  strength. 
It  would  be  more  convenient  to  express  the  circular 
pitch  in  whole  inches,  and  the  most  natural  divisions 

an  inch>  as  !"  F»  3"  P/>  4"  P'»  and  so  OD-      But  as 
circumference  of  the  pitch  circle  must  contain  tho 

iteh    some    whole   number    of   times,    corro 

tO    tne     numker     of     teetn     m     tne     gearr   tlie 

diameter  of  the  pitch  circle  will  often  be  of  a  size  not 
readily  measured  with  a  common  rule.  This  is  because 
the  circumference  of  a  circle  is  equal  to  3.1416  times 
the  diameter,  or  the  diameter  is  equal  to  the  circumfer- 
ence multiplied  by  .3183. 
Pitch,  in  In  practice,  it  is  better  that  the  diameter  should  be 

Terms    of  the 

Diameter.  of  some  size  conveniently  measured.  The  same  applies 
to  the  distance  between  centers.  Hence  it  is  generally 
more  convenient  to  assume  the  pitch  in  terms  of  the 
diameter.  In  Chapter  II.  was  given  a  definition  of  the 
module,  and  also  how  to  obtain  the  module  from  the 
circular  pitch. 

an!irCalaDiameh  ^e  can  a^so  assume  ^ne  module  and  pass  to  its  equiv- 
terPitcb.  alent  circular  pitch.  If  the  circumference  of  the  pitch 
circle  is  divided  by  the  number  of  teeth  in  the  gear, 
the  quotient  will  be  the  circular  pitch.  In  the 
same  manner,  if  the  diameter  of  the  pitch  circle  is 
divided  by  the  number  of  teeth,  the  quotient  will 
be  the  module.  Thus,  if  a  gear  is  12  inches  pitch 
diameter  and  has  48  teeth,  dividing  12"  by  48,  the 
quotient  J"  is  the  module  of  this  gear.  In  prac- 


PROVIDENCE,    R.    I.  17 

tice,  the  module  is  taken   in   some  convenient  part  of 

an  inch,  as  •£"  module  and  so  on.     It  is  convenient  in   AV^T1^-011 

of  Module  Dia- 

calculation  to  designate  one  of  these  modules  by  s,  as  meter  pitch- 

in  Chapter  II.     Thus,  for  -J-"  module,  s  is  equal  to  £". 

Generally,  in  speaking  of  the  module,  the  denominator 

of  the  fraction  only  is  named.     -J-"  module  is  then  called 

3  diametral  pitch.     That  is,  it  has  been  found  more 

convenient  to  take  the  reciprocal  of  the  module  in  mak- 

ing calculation.     The  reciprocal  of  a  number  is  1  divi-    Reciprocal  of 

ded  by  that  number.      Thus  the  reciprocal  of  |-  is  4, 

because  \  goes  into  1  four  times. 

Hence,  we  come  to  the  common  definition  : 

DIAMETRAL  PITCH  is  the  number  of  teeth  to  one  inch    Diametral 
of  diameter  of  pitch  circle.     Let  this  be  denoted  by  P.  Pl 
Thus,  y  diameter  pitch  we  would  call  4  diametral  pitch 
or  4  P,  because  there  would  be  4  teeth  to  every  inch  in 
the    diameter  of  pitch   circle.     The  circular  pitch  and 
the  different   parts  of  the  teeth   are  derived  from  the 
diametral  pitch  as  follows. 

^^  =  P',  or  3.1416  divided  by  the  diametral  pitch 
is  equal  to  the  circular  pitch.  Thus  to  obtain  the  cir- 
cular  for  4  diametral  pitch,  we  divide  3.1416  by  4  and  To  obtain  Cir. 
obtain  .7854  for  the  circular  pitch,  corresponding  to;^-f?omrDiinie! 
diametral  pitch.  trai  Pitch. 

In  this  case  we  would  write  P=4,  P'=.7854",  s  =  y. 
~  =s,  or  one  inch  divided  by  the  number  of  teeth  to  an 
inch,  gives  distance  on  diameter  of  pitch  circle  occupied 
by  one  tooth  or  the  module.  The  addendum  or  face  of 
tooth  is  the  same  distance  as  the  module. 

g  ==  P,  or  one  inch  divided  by  the  module  equals  num- 
ber of  teeth  to  one  inch  or  the  diametral  pitch. 

i^-  =  £,  or  1.57  divided  by  the  diametral  pitch  gives  ametmi'pitchto 

find  the  Thick- 

thickness  of  tooth  at  pitch   line.     Thus,  thickness   of  ness  of  Tooth 

at  the    Pitch 

teeth  along  the  pitch  line  for  4  diametral  pitch  is  .392".  Line. 


=  D',  or  number  of  teeth  in  a  gear  divided  by  the 
diametral  pitch  equals  diameter  of  the  pitch  circle. 
Thus  for  a  wheel,  60  teeth,  12  P,  the  diameter  of  pitch 
circle  will  be  5  inches.  <frrcleof  Pitch 

£if  =  D,  or  add  2  to  the  number  of  teeth  in  a  wheel    Given,  the 

P  Number  of 

and   divide  the  sum    by  the  diametral  pitch  ;    and  the  Teethinawheel 

J  and  the  Diame- 

tral Pitch  to 
find  the  Whole 
Diameter. 


18  BROWN   &    SHAEPE    MFG.    CO. 

quotient  will  be  the  whole  diameter  of  the  gear  or  the 
diameter  of  the  addendum  circle.  Thus,  for  60  teeth, 
12  P,  the  diameter  of  gear  blank  will  be  5T2g-  inches. 

D,=P,  or  number  of  teeth  divided  by  diameter  of 
pitch  circle  in  inches,  gives  the  diametral  pitch  or 
number  of  teeth  to  one  inch.  Thus,  in  a  wheel,  24 
teeth,  3  inches  pitch  diameter,  the  diametral  pitch  is  8. 
^^=P,  or  add  2  to  the  number  of  teeth;  divide  the 
sum  by  the  whole  diameter  of  gear,  and  the  quotient 
will  be  the  diametral  pitch.  Thus,  for  a  wheel  3^" 
diameter,  14  teeth,  the  diametral  pitch  is  5. 

D'  P=N,  or  diameter  of  pitch  circle,  multiplied  by 
diametral  pitch  equals  number  of  teeth  in  the  gear. 
Thus,  in  a  gear,  5  pitch,  8"  pitch  diameter,  the  num- 
ber of  teeth  is  40. 

DP — 2=N  or  multiply  the  whole  diameter  of  the 
gear  by  the  diametral  pitch,subtract  2,  and  the  remain- 
der will  be  the  number  of  teeth. 

j^2  =  s,  or  divide  the  whole  diameter  of  a  spur  gear 
by   the   number  of  teeth  plus   two,  and  the  quotient 
will  be  the  module. 

rraihpitchiame  When  we  say  the  diametrical  pilch  we  shall  mean  the 
number  of  teeth  to  one  inch  of  diameter  of  pitch  cir- 
cle, or  P,  (V=P). 

When  we  say  the  diametral  pitch  we 
shall  mean  the  number  of  teeth  to  one  inch  of  diameter 
of  pitch  circle,  or  P,  (7=?). 

am?t£aitaipi£h     When  the  circular  pitch  is  given,  to  find  the  corre- 

ptteL  circular  spending  diametral  pitch,  divide  3.1416  by  the  circular 
pitch.  Thus  1.57  P  is  the  diametral  pitch  correspond- 
ing to  2-inch  circular  pitch,  (^i/jM— P). 

Example.  What   diametral  pitch   corresponds  to  ^"   circular 

pitch  ?  Remembering  that  to  divide  by  a  fraction  we 
multiply  by  the  denominator  and  divide  by  the  numer- 
ator, we  obtain  6.28  as  the  quotient  of  3.1416  divided  by 
J  .  6.28  P,  then,  is  the  diametral  pitch  corresponding 
to  ^  circular  pitch.  This  means  that  in  a  gear  of  ^ 
inch  circular  pitch  there  are  six  and  twenty-eight  one 
hundredths  teeth  to  every  inch  in  the  diameter  of  the 
pitch  circle.  In  the  table  of  tooth  parts  the  diametral 


PROVIDENCE,   R.    I.  19 

pitches  corresponding  to  circular  pitches  are  carried 
out  to  four  places  of  decimals,  but  in  practice  three 
places  of  decimals  are  enough. 

When  two  gears  are  in  mesh,  so  that  their  pitch 
circles  just  touch,  the  distance  between  their  axes  or 
centers  is  equal  to  the  sum  of  the  radii  of  the  two  gears. 
The  number  of  the  modules  between  centers  is  equal  to 
half  the  sum  of  number  of  teeth  in  both  gears.  This 
principle  is  the  same  as  given  in  Chapter  II..  page  6,  Rule  to  find 

Distance  be- 

but  when  the  diametral   pitch  and  numbers  of  teeth  in  tween  centers, 
two  gears  are  given,  add  together  the  numbers  of  teeth  in 
the  two  wheels  and  divide  half  the  sum  by  the  diametral 
pitch.      The  quotient  is  the  center  distance. 

A  gear  of  20  teeth,  4  P,  meshes  with  a  gear  of  50  Example, 
teeth  ;    what  is  the  distance  between  their  axes  or  cen- 
ters?   Adding  50  to  20  and  dividing  half  the  sum  by  4, 
we  obtain  8J"  as  the  center  distance. 

The  term  diametral  pitch  is  also  applied  to  a  rack. 
Thus,  a  rack  3  P,  means  a  rack  that  will  mesh  with  a 
gear  of  3  diametral  pitch. 

It  will  be  seen  that  if  the  expression  for  the  module    Fractional 
has  any  number  except  1  for  a  numerator,  we  cannot  pitch.™ 
express  the  diametral  pitch  by  naming  the  denominator 
only,     Thus,  if   the   addendum    or  module   is  T4^,  the 
diametral   pitch  will   be    21,  because  1  divided  by  T4¥ 
equals  2£. 

The  term  module  is  much  used  where  gears  are  made 
to  metric  sizes,  for  the  reason  that,  the  millimeter  being 
so  short,  the  module  is  conveniently  expressed  in  milli- 
meters. If  we  know  the  module  of  a  gear  we  can  figure 
the  other  parts  as  easily  as  we  can  if  we  know  either 
the  circular  pitch  or  the  diametral  pitch.  The  module 
is,  in  a  sense,  an  actual  distance,  while  the  diametral 
pitch,  or  the  number  of  teeth  to  an  inch,  is  a  relation  or 
merely  a  ratio.  The  meaning  of  the  module  is  not 
easily  mistaken. 


20 


CHAPTER  VI. 

SINGLE-CURVE  GEARS  HAVING  LESS  THAN  30  TEETH— GEARS  ABD 
RACKS  TO  MESH  WITH  GEARS  HAVING  LESS  THAN  30  TEETH. 


^n  Fig-  10>  tt16  construction  of  the  rack  is  the  same 
as  the  construction  of  the  rack  in  Chapter  IV.  The 
gear  in  Fig.  10  is  drawn  from  base  circle  out  to  adden- 
dum circle,  by  the  same  method  as  the  gear  in  Chapter 
III.,  but  the  spaces  inside  of  base  circle  are  drawn  as 
follows : 
Flanks  of  In  gears,  12  to  19  teeth,  the  sides  of  space  inside 

Gears  m  low 

Numbers  of  of  the  base  circle  are  radial  for  a  distance,  a  b,  equal 
to  JJ,  or  3.5  divided  by  the  product  of  the  pitch  by  the 
number  of  teeth.  In  gears  with  more  than  19  teelh 
the  radial  construction  is  omitted. 

construction     Then,  with   one  leg  of  dividers  in  pitch  circle  in 
tinued.  center  of  next  tooth,  e,  and  other  leg  just   touching 

one  of  the  radial  lines  at  b,  continue  the  tooth  side 
into  c,  until  it  will  touch  a  fillet  arc,  whose  radius  is 
1  the  width  of  space  at  the  addendum  circle.  The 
part,  br  c'9  is  an  arc  from  center  of  tooth  g,  etc.  The 
flanks  of  teeth  or  spaces  in  gear,  Fig.  11,  are  made  the 
same  as  those  in  Fig.  10. 

This  rule  is  merely  conventional  or  not  founded 
upon  any  principle  other  than  the  judgment  of  the  de- 
signer, to  effect  the  object  to  have  spaces  as  wide  as 
practicable,  just  below  or  inside  of  base  circle,  and 
then  strengthen  flank  with  as  large  a  fillet  as  will  clear 
addenda  of  any  gear.  If  flanks  in  any  gear  will  clear 
addenda  of  a  rack,  they  will  clear  addenda  of  any 
Internal  Gear,  other  gear,  except  internal  gears.  An  internal  gear  is 
one  having  teeth  upon  the  inner  side  of  a  rim  or  ring. 
Now,  it  will  be  seen  that  the  gear,  Fig.  10,  has  teeth 


PEOVIDENCE,   R.  I. 


Fig.  10 


22  BKOWN    &    SHAEPE    MFG.    CO. 

too  much  rounded  at  the  points  or  at  the  addendum 
circle.     In  gears  of  pitch  coarser  than  10  to  inch  (10 
Adduenninf  of  P)'    and    having   less   than  30   teeth,    this   rounding 
Teeth.  becomes  objectionable.    This  rounding  occurs,  because 

in  these  gears  arcs  of  circles  depart  too  far  from  the 
true    involute  curve,    being  so  much  that  points   of 
teeth  get  no  bearing  on  flanks  of  teeth  in  other  wheels. 
In  gear,  Fig.  11,  the  teeth  outside  of  base  circle  are 
made  as  nearly  true  involute  as  a  workman  will  be  able 
to  get  without  special  machinery.  This  is  accomplished 
tio^PtorTmei5- as  * °^ows :  draw  three  or  four  tangents  to  the  base 
volute.  circle,  i  if,  jj',  k  k ',  1 1',  letting  the  points  of  tangency 

on  base  circle  i',j',  k\  I'  be  about  \  or  J  the  circular  pitch 
apart ;  the  first  point,  i',  being  distant  from  i,  equal  to 
\  the  radius  of  pitch  circle.  With  dividers  set  to  J 
the  radius  of  pitch  circle,  placing  one  leg  in  i'9  draw 
the  arc,  a'  i  j;  with  one  leg  in  j',  and  radius  j'  j, 
draw/  k;  with  one  leg  in  k',  and  radius  k'  k  draw  k  L 
Should  the  addendum  circle  be  outside  of  I,  the  tooth 
side  can  be  completed  with  the  last  radius,  I'  L  The 
arcs,  a'  ij,  j  k  and  k  I,  together  form  a  very  close 
approximation  to  a  true  involute  from  the  base  circle, 
i'  j'  k'  I'.  The  exact  involute  for  gear  teeth  is  the 
curve  made  by  the  end  of  a  band  when  unwound  from 
a  cylinder  of  the  same  diameter  as  base  circle. 

The   foregoing    operation  of   drawing  tooth  sides, 
although  tedious  in  description,  is  very  easy  of  practical 
application. 
Rounding    of      It  will  also  be  seen  that  the  addenda  of  rack  teeth 

Addenda  of 

Rack.  in  Fig.  10,  interfere  with  the  gear-teeth  flanks,  as  aL 

m  n;  to  avoid  this  interference,  the  teeth  of  rack,  Fig. 
11,  are  rounded  at  points  or  addenda. 

It  is  also  necessary  to  round  off  the  points  of  invo- 
lute teeth  in  high -numbered  gears,  when  they  are  to 
interchange  with  low -numbered  gears.    In  interchange- 
able sets  of  gears  the  lowest-numbered  pinion  is  usual- 
Tempietsly  12.     Just  how  much  to  round  off  can  be  learned  by 

necessary  for   * 

Rounding  off  makingf  templets  of   a  few  teeth  out  of  thin  metal   or 

Points  of  teeth.  fc  „ T         ,,  -,  i  , 

cardboard,  for  the  gear  and  rack,  or,  two  gears  re- 
quired, and  fitting  addenda  of  teeth  to  clear  flanks. 
However  accurate  we  may  make  a  diagram,  it  is  quite 


PROVIDENCE,  E.  I. 


SINGLE  CURVE  GEAR,  2  P.,  12  TEETH 
IN  MESH  WITH   RACK. 


P=2 
N—  12 

P'=1.57" 

t=.7854" 
.500" 
D"-  1.000" 

s  +  f  =.5785" 
1.078 
D'=6." 
D  = 


Fig.  11 


BROWN    &    SHARPE    MFG      CO. 

as  well   to  make  templets  in  order  to  shape  cutters 
accurately. 

a  DseTorcu°t-      lt  is  best  to  make  cutters  to  corrected  diagrams,  as 

in  Fig.  11.     When  corrected  diagrams  are  made,  as 
in  Fig.  1 1 ,  take  the  following  : 

For  12  and  13  teeth,  diagram  of  12  teeth. 

44     14    to    16       "            «         «  14  u 

44     17     "     20       44            "         "   17  tt 

44     21     4-     25       »            "         tt   21  it 

44     26     4<     34       "            tt         u  26  4' 

44     35     "     54       4'            *t         n  35  n 

44     55     4t   134       "            .t         it   55  u 

4'   135    44rack,     "           "         *'135  4' 
Templets  for  large  gears  must  be  fitted  to  run  with 
12  teeth. 


25 


CHAPTER  VII. 
DOUBLE-CURVE  TEETH— GEAR,  15  TEETH— RACK. 


In  double-curve  teeth  the  formation  of  tooth  sides 
changes  at  the  pitch  line.-  In  all  gears  the  part  of  Faces  are  con- 
teeth  outside  of  pitch  line  is  convex ;  in  some  gears 
the  sides  of  teeth  inside  pitch  line  are  convex ;  in  some, 
radial ;  in  others,  concave.  Convex  faces  and  concave 
flanks  are  most  familiar  to  mechanics.  In  interchange- 
able sets  of  gears,  one  gear  in  each  set,  or  of  each 
pitch,  has  radial  flanks.  In  the  bast  practice,  this  gear 
has  fifteen  teeth.  Gears  with  more  than  fifteen  teeth, 
have  concave  flanks ;  gears  with  less  than  fifteen  teeth, 
have  convex  flanks.  Fifteen  teeth  is  called  the  Base 
of  this  system. 

We  will  first  draw  a  gear  of  fifteen  teeth.      This  of  Construction 
fifteen-tooth   construction   enters   into   gears  of   any 
number  of  teeth  and  also  into  racks.     Let  the  gear  be 
3  P.     Having  obtained  data,  we  proceed  as  follows : 

1.  Draw  pitch  circle  and  point  it  off  into  parts  equal 
to  one-thirtieth  of  the  circumference,  or  equal  to  thick- 
ness of  tooth  —  t. 

2.  From  the  center,  through  one  of  these  points,  as 
at  T,  Fig.  12,  draw  line  OTA.     Draw  addendum  and 
whole-depth  circles. 

3.  About  this  point,  T,  with  same  radius  as  15-tooth 
pitch  circle,  describe  arcs  A  K  and  O  k.    For  any  other 
double-curve  gear  of  3  P.,  the  radius  of  arcs,  A  K  and 
O  &,  will  be  the  same  as  in  this  15-tooth  gear=2J". 
In  a  15-tooth  gear,  the  arc,  O  k,  passes  through  the 
center  O,  but  for  a  gear  having  any  other  number  of 
teeth,   this  construction   arc  does   not  pass   through 
center  of  gear.    Of  course,  the  15-tooth  radius  of  arcs, 
A  K  and  O  k,  is  always  taken  from  the  pitch  we  are 
working  with. 


26 


BROWN    &    SHARPE   MFG.    CO. 


GEAR,  3  P.,  15  TEETH 


N  =  15 

P'=  1.0472" 

t=    .5236" 

S  =    .3333" 

D"=    .6666" 

s+f  =    .3S67" 

D"+/=    .7190" 

D'=  5.0000" 

D  =  5.6666" 


JEPIg.  IS. 

DOUBLE    CURVE    GEAR. 


PROVIDENCE,    R.    I. 

4.  Upon  these  arcs  on  opposite  sides  of  line    OTA, 
Jay  off  tooth  thickness,  A  K  and  O  k,  and  draw  line 
KT&. 

5.  Perpendicular  to  K  T  &,  draw  line  of  pressure, 
L  T  P ;  also  through  O  and  A,  draw  lines  A  R  and  O  r, 
perpendicular  to  K  T  k.    The  line  of  pressure  is  at 
an  angle  of  78°  with  the  radius  of  gear. 

6.  From  O,  draw  a  line  O  R  to  intersection  of  A  R 
with  K  T  &.     Through  point  c,  where  O  R  intersects 
L  P,  describe  a  circle  about  the  center,  O.     In  this 
circle  one  leg  of  dividers  is  placed  to  describe  tooth 
faces 

7.  The  radius,    c  d,   of  arc  of  tooth  faces   is   th*. 
straight  distance  from  c  to  tooth-thickness  point,  5, 
on  the  other  side  of  radius,  O  T.     With  this  radius,  c  b, 
describe  both  sides  of  tooth  faces. 

8.  Di-aw  flanks  of  all  teeth  radial,  as  O  e  and  O  f. 
The  base  gear,  15  teeth~oidy,  has  radial  flanks. 

9.  With  radius  equal  to  one-seventh  of  the  widest 
part  of  space,  as  g  h,  draw  fillets  at  bottom  of  teeth. 

The  foregoing  is  a  close  approximation  to  epicy-  ti 
cloidal  teeth.  To  get  exact  teeth,  make  two  1 5 -tooth cioidai  Teeth. 
gears  of  thin  metal.  Make  addenda  long  enough  to 
come  to  a  point,  as  at  n  and  q.  Make  radial  flanks,  as 
at  m  and  p,  deep  enough  to  clear  addenda  when  gears 
are  in  mesh.  First  finish  the  flanks,  then  fit  the  long 
addenda  to  the  flanks  when  gears  are  in  mesh. 

When  these  two  templet  gears  are  alike,  the  centers  T ^^ sd  a  r  d 
are  the  right  distance  apart   and  the  teeth  interlock 
without  backlash,  they  are  exact.     One  of  these  tem- 
plet gears  can  now  be  used  to  test  any  other  templet 
gear  of  the  same  pitch. 

Gears  and  racks  will  be  right  when  they  run  cor- 
rectly with  one  of  these  1 5-tooth  templet  gears.  Five 
or  six  teeth  are  enough  to  make  in  a  gear  templet. 

DOUBLE  CURVE  RACK.— Let  us  draw  a  rack  3  P. 
Having  obtained  data  of  teeth  we  proceed  as  follows : 

1.  Draw  pitch  line  and  point  it  off  in  parts  equal 
to  one- half  the  circular  pitch.     Draw  addendum  and 
whole-depth  lines. 

2.  Through  one  of  the  points,  as  at  T,  Fig.  13,  draw 
line  OTA  perpendicular  to  pitch  line  of  rack. 


28 


BROWN    &    SHARPE    MFG.    CO. 


RACK.3  P. 
P=  3 

P'=  1.0472" 

t  =    .5236" 

8  =    .3333" 

D'=    .6666" 

=     .3857" 

=    .7190" 


.  13 

DOUBLE  CURVE    RACK. 


PROVIDENCE,    R.    I. 

3.  About  T  make  precisely  the  same  construction  as 
was  made  about  T  in  Fig.  12.     That  is,  with  radius  of 
15-tooth  pitch  circle  and  center  T  draw  arcs  O  k  and 
A  K ;   make    O  k  and  A  K  equal  to  tooth  thickness  ; 
draw  K  T  k  •  draw  O  r,  A  R,  and  line  of  pressure,  each 
perpendicular  to  K  T  k. 

4.  Through  E  and  r,  draw  lines  parallel  to  O  A. 
Through  intersections  c  and  c'  of  these  lines,  with 
pressure  line  L  P,  draw  lines  parallel  to  pitch  line. 

5.  In  these  last  lines  place  leg  of  dividers,  and  draw 
faces  and  flanks  of  teeth  as  in  sketch. 

6.  The  radius  c'  d'  of  rack-tooth  faces  is  the  same 
length  as  radius  c  d  of  rack-tooth  flanks,  and  is  the 
straight  distance  from  c  to  tooth-thickness  point  b  on 
opposite  side  of  line  O  A. 

7.  The  radius  for  fillet  at  bottom  of  rack  teeth  is 
equal  to  j-  of  the  widest  part  of  tooth  space.     This 
radius   can   be   varied   to    suit  the  judgment   of  the 
designer,  so  long  as  a  fillet  does  not  interfere  with 
teeth  of  engaging  gear. 


Fig.  14 


Racks  of  the  same  pitch,  to  mesh  with  interchange- 
able gears,  should  be  alike  when  placed  side  by  side, 
and  fit  each  other  when  placed  together  as  in  Fig.  14. 

In  Fig.  13,  a  few  teeth  of  a  15-tooth  wheel  are  shown 
in  mesh  with  the  rack. 


30 


CHAPTER  VIII. 

DOUBLE-CURVE  SPUR  GEARS,  HAVING  MORE  AND  FEWER  THAN 
15  TEETH— ANNULAR  GEARS, 


^construction  Let  us  <jraw  two  gears,  12  and  24  teeth,  4  P,  in 
mesh.  In  Fig.  15  the  construction  lines  of  the  lower 
or  24-tooth  gear  are  full.  The  upper  or  12-tooth  gear 
construction  lines  are  dotted.  The  line  of  pressure, 
L  P,  and  the  line  K  T  k  answer  for  both  gears.  The 
arcs  A  K  and  O  k,  are  described  about  T.  The  radius 
of  these  arcs  is  the  radius  of  pitch  circle  of  a  gear  15 
teeth  4  pitch.  The  length  of  arcs  A  K  and  O  Jc  is  the 
tooth  thickness  for  4  P.  The  line  K  T  k  is  obtained 
the  same  as  in  Chapter  VII.  for  all  double-curve  gears, 
the  distances  only  varying  as  the  pitch.  Having  drawn 
the  pitch  circles,  the  line  K  T  /£,  and,  perpendicular  to 
K  T  /£,  the  lines  A  R,  O  r  and  the  line  of  pressure 
L  T  P,  we  proceed  with  the  24-tooth  gear  as  follows : 

1.  From  center  C,  through  r,  draw  line  intersecting 
line  of  pressure  in  m.     Also  draw  line  from  center  C 
to  B,  crossing  the  line  of  pressure  L  P  at  c. 

2.  Through  m  describe  circle  concentric  with  pitch 
circle  about  C.     This  is  the  circle  in  which  to  place 
one  leg  of  dividers  to  describe  flanks  of  teeth. 

3.  The  radius,  m  n,  of  flanks  is  the  straight  distance 
from'm  to  the  first  tooth-thickness  point  on  other  side 
of  line  of  centers,  C  C',  at  v.     The  arc  is  continued  to 
n,  to  show  how  constructed.     This  method  of  obtain- 
ing radius  of  double-curve  tooth  flanks  applies  to  all 
gears  having  more  than  fifteen  teeth. 

4.  The  construction  of  tooth  faces  is  similar  to  15- 
tooth  wheel  in  Chapter  VII.     That  is  :    Draw  a  circle 
through   c    concentric  to   pitch  circle  ;  in  this  circle 
place  one  leg   of   dividers  to   draw  tooth  faces,  the 
radius  of  tooth  faces  being  c  b. 


PROVIDENCE,    R.    I. 


PINION,  12  TEETH, 
GEAR  24  TEETH,  .4  P. 

P=4 

N  =12  and  24 

P'=  .7854" 

t  =  .3927" 

8  =  .2500" 

D^=  .5000' 

St/  =  .2893' 

0" +/ =  .5393" 


.  15 


PINION 


GEAR 


DOUBLE  CURVE  GEARS  IN   MESH. 


32  BROWN   &    SHARPE    MFG.    CO. 


of  Sff  ^"SS!  5-  -The  radius  of  fillets  at  roots  of  teeth  is  equal  to 
one-seventh  the  width  of  space  at  addendum  circle. 

Flanks  for  12,     The   constructions  for  flanks  of  12,  13  and  14 

13  and  14  Teeth.'  teeth  are  similar  to  each  other  and  as  follows  : 

1.  Through  center,  C',  draw  line  from  K,  intersecting 
line  of  pressure  in  u.     Through  u  draw  circle  about 
C'.     In  this  circle  one  leg  of  dividers  is  placed  for 
drawing  flanks. 

2.  The  radius  of  flanks  is  the  distance  from  u  to 
the   first  tooth-thickness  point,  e,  on  the  same  side  of 
C  T  C'.     This  gives  convex  flanks.     The  arc  is  con- 
tinued to  V,  to  show  construction. 

3.  This  arc  for  flanks  is  continued  in  or  toward  the 
center,  only  about  one  sixth  of  the  working  depth  (or 
£  s.)  ;  the  lower  part  of  flank  is  similar  to  flanks  of 
gear  in  Chapter  VI. 

4.  The  faces  are  similar  to  those  in  15-  tooth  gear, 
Chapter  VII.,  and  to  the  24-tooth  gear  in  the  fore- 
going, the  radius  being  w  y  ;  the  arc  is  continued  to  xt 
to  show  construction. 

Annular  Gears.  ANNULAR  GfiAns.  Gears  with  teeth  inside  of  a  rim 
or  ring  are  called  Annular  or  Internal  Gears.  The 
construction  of  tooth  outlines  is  similar  to  the  fore- 
going, but  the  spaces  of  a  spur  external  gear  become 
the  teeth  of  an  annular  gear. 

Prof.  MacCord  has  shown  that  in  the  system  just 
described,  the  pinion  meshing  with  an  annular  gear, 
must  differ  from  it  by  at  least  fifteen  teeth.  Thus, 
a  gear  of  24  teeth  cannot  work  with  an  annular  gear 
of  36  teeth,  but  it  will  work  with  annular  gears  of  39 
teeth  and  more.  The  fillets  at  the  roots  of  the  teeth 
mnst  be  of  less  radius  than  in  ordinary  spur  gears.  An 
annular  gear  differing  from  its  mate  by  less  thn.n  15 
teeth  can  be  made.  This  will  be  shown  in  Part  II. 

Annular-gear  patterns  require  more  clearance  for 
moulding  than  external  or  spur  gears. 

Pinions.  In  speaking    of  different-sized  gears,  the    smallest 

ones  are  often  called  "  pinions." 

The  angle  of  pressure  in  all  gears  except  involute, 
constantly  changes.  78°  is  the  pressure  angle  in 
double-curve,  or  epicycloidal  gears  for  an  instant 


PROVIDENCE.   K.   I.  33 

only;  in  our  example,  it  is  78°  when  one  side  of  a 
tooth  reaches  the  line  of  centers,  and  the  pressure 
against  teeth  is  applied  in  the  direction  of  the  arrows. 

The  pressure  angle  of  involute  gears  does  not 
change.  An  explanation  of  the  term  angle  of  pressure 
is  given  in  Part  II. 

We  obtain  the  forms  for  epicycloidal  gear  cutters 
by  means  of  a  machine  called  the  Odontom  Engine. 
This  machine  will  cut  original  gears  with  theoretical 
accuracy. 

It  has  been  thought  best  to  make  24  gear  cutters    24  Double- 
curve.  Gear 

for  each  pitch.     This  enables  us  to  fill  any  require-  ^hup^  for 
ment  of  gear-cutting  very  closely,  as  the  range  covered 
by  any  one  cutter  is  so  small  that  it  is  exceedingly  near 
to  the  exact  shape  of  all  gears  so  covered. 

Of  course,  a  cutter  can  be  exactly  right  for  only  one 
gear.  Special  cutters  can  be  made,  if  desired. 


34 


CHAPTER   IX. 

BEVEL-GEAR  BLANKS, 


Bevel  Gears  connect  shafts  whose  axes  meet  when 
Bevel6  Gears*  sufficiently  prolonged.  The  teeth  of  bevel  gears  are 
formed  upon  formed  about  the  frustrums  of  cones  whose  apexes 

frustrums    of 

cones.  are  at  the  same  point  where  the  shafts  meet.     In  Fig. 

16  we  have  the  axes  A  O  and  B  O,  meeting  at  O,  and 
the  apexes  of  the  cones  also  at  O.  These  cones  are 
called  the  pitch  cones,  because  they  roll  upon  each 
other,  and  because  upon  them  the  teeth  are  pitched. 
If,  in  any  bevel  gear,  the  teeth  were  sufficiently  pro- 
longed toward  the  apex,  they  would  become  infinitely 
small ;  that  is,  the  teeth  would  all  end  in  a  point,  or 
vanish  at  O.  We  can  also  consider  a  bevel  gear  as 
beginning  at  the  apex  and  becoming  larger  and  larger 
as  we  go  away  from  the  apex.  Hence,  as  the  bevel 
gear  teeth  are  tapering  from  end  to  end,  we  may  say 


BEVEL  GEAR  PITCH  CONES. 


Fig.    16. 


that  a  bevel  gear  has  a  number  of  pitches  and  pitch 
circles,  or  diameters  :  in  speaking  of  the  pitch  of  a 
bevel  gear,  we  mean  always  the  pitch  at  the  largest 


PROVIDENCE,    R.    I.  35 

pitch  circle,  or  at  the  largest  pitch  diameter,   as  at 
bd,  Fig.  17. 

Fig.  17  is  a  section  of  three  bevel  gears,  the  gear 
o  B  q  being  twice  as  large  as  the  two  others.  The 
outer  surface  of  a  tooth  as  m  m'  is  called  the  face  of  Construction 

ot  Bevel  Gear 

the  tooth.      The  distance  m  in'  is    usually  called  the  Blanks, 
length  of  the  face  of  the  tooth,  though  the  real  length 
is  the  distance  that  it  occupies  upon  the  line  O  i.     The 
outer  part  of  a  tooth  at  m  n  is  called  its  large  end,  and 
the  inner  part  m'  n'  the  small  end. 

Almost  all  bevel  gears  connect  shafts  that  are  at 
right  angles  with  each  other,  and  unless  stated  other- 
wise we  always  understand  that  they  are  so  wanted. 

The  directions  given  in  connection  with  Fig.  17 
apply  to  gears  with  axes  at  right  angles. 

Having  decided  upon  the  pitch  and  the  numbers  of 
teeth  :— 

1.  Draw  centre  lines  of  shafts,  A  O  B  and  COD, 
at  right  angles. 

2.  Parallel  to  A  O  B,  draw  lines  a  b  and  c  d,  each 
distant  from  A  O  B,   equal  to  half  the   largest  pitch 
diameter  of  one  gear.     For  24  teeth,  4  pitch,  this  half 
largest  pitch  diameter  is  B". 

3.  Parallel  to  COD,  draw  lines  e  f  and  g  h,  dis- 
tant from  COD,  equal   to   half   the   largest   pitch 
diameter  of  the  other  gear.     For  a  gear,  12  teeth,  4 
pitch,  this  half  largest  pitch  diameter  is  1|". 

4.  At  the  intersection  of  these  four  lines,  draw 
lines  O  i,  O  j,  O  k,  and  O  1 ;  these  lines  give  the  size 
and  shape  of  pitch  cones.     We  call  them  u  Cone  Pitch 
Lines." 

5.  Perpendicular  to  the  cone-pitch  lines  and  through 
the  intersection  of  lines  a  b,  c  d,  e  f,   and  g  h,   draw 
lines  m  n,  o  p,  q  r.     We  have  drawn  also  u  v  to  show 
that  another  gear  can  be  drawn  from  the  same  diagram. 
Four  gears,  two  of  each  size,  can  be  drawn  from  this 
diagram. 

6.  Upon  the  lines  m  n,  o  p,  q  r,  the    addenda  and 
depth  of .  the  teeth  are  laid  off,  these    lines  passing 


36 


BROWN    &   SHARPE   MFG.    CO. 

through  the  largest  pitch  circle  of  the  gears.  Lay  off 
the  addendum,  it  being  in  these  gears  £".  This  gives 
distance  m  u,  o  p,  q  r,  and  u  v  equal  to  the  working 
depth  of  teeth,  which  in  these  gears  is  y.  The 
addendum  of  course  is  measured  perpendicularly  from 
the  cone  pitch  lines  as  at  k  r. 

7.  Draw   lines   O  m,   O  n,    O  p,    O  o,    O  q,    O  r. 
These  lines  give  the  height  of  teeth  above  the  cone- 
pitch  lines    as   they   approach  O,  and  would  vanish 
entirely  at  O.     It  is  quite   as  well  never  to  have  the 
length  of  teeth,  or  face,  m  m'  longer  than  one-third 
the  apex  distance  m  O,  nor  more  than  two  and  one- 
half  times  the  circular  pitch. 

8.  Having  decided  upon  the  length   of  face,  draw 
limiting  lines  m'n'  perpendicular  to  i  O,  q'  r'  perpen- 
dicular to  k  O,  and  so  on. 

The  distance  between  the  cone-pitch  lines  at  the 
inner  ends  of  the  teeth  m'n'  and  q'  r'  is  called  the  inner 
or  smaller  pitch  diameter,  and  the  circle  at  these  points 
is  called  the  smallest  pitch  circle.  We  now  have  the 
outline  of  a  section  of  the  gears  through  their  axes. 
The  distance  m  r  is  the  whole  diameter  of  the  pinion. 
The  distance  q  o  is  the  whole  diameter  of  the  gear. 
Gear  In  practice  these  diameters  can  be  obtained  by  measur- 


3ieH s ur i S y  ™%  ^e  drawing.  The  diameter  of  pinion  is  3.45"  and 
Drawings.  of  the  gear  6.22".  *  We  can  find  the  angles  also  by 
measuring  the  drawing  with  a  protractor.  In  the 
absence  of  a  protractor,  templetes  can  be  cut  to  the 
drawing.  The  angle  formed  by  line  m  m'  with  a  b  is 
the  angle  of  face  of  pinion,  in  this  pinion  59°  11',  or 
59^°  nearly.  The  lines  q  q'  and  g  h  give  us  angle  of 
face  of  gear,  for  this  gear  22°  19',  or  22J°  nearly 
The  angle  formed  by  m  n  with  a  b  is  called  the  angle 
of  edge  of  pinion,  in  our  sketch  26°  34',  or  about  26|°. 
The  angle  of  edge  of  gear,  line  q  r  with  g  h,  is  63°  20', 
or  about  63 1°.  In  turning  blanks  to  these  angles  we 
place  one  arm  of  the  protractor  or  templet  against  the 
end  of  the  hub,  when  trying  angles  of  a  blank.  Some 
designers  give  the  angles  from  the  axes  of  gears,  but 


PROVIDENCE,    R.    I. 


37 


BROWN    &   SHAEPE   MFG.    CO., 

it  is  not  convenient  to  try  blanks  in  this  way.  The 
method  that  we  have  given  comes  right  also  for  angles 
as  figured  in  compound  rests. 

When  axes  are  at  right  angles,  the  sum  of  angles 
of  edge  in  the  two  gears  equals  90°,  and  the  sums  of 
angle  of  edge  and  face  in  each  gear  are  alike. 

The  angles  of  the  axes  remaining  the  same,  all  pairs 
of  bevel  gears  of  the  same  ratio  have   the   same  angle 
of  edge  ;  all  pairs  of  same  ratio  and  of  same  numbers 
of  teeth  have  the  same  angles  of  both  edges  and  faces 
independent  of  the  pitch.      Thus,  in  all  pairs  of  bevel 
gears  having  one  gear  twice  as  large  as  the  other,  with 
axes  at  right  angles,  the  angle  of   edge  of  large  gear 
is  63°  26',  and  the  angle  of  edge  of  small  gear  is  26°  34'. 
In  all  pairs  of  bevel  gears  with  axes  at  right  angles, 
one  gear  having  24  teeth  and  the  other  gear  having  1 2 
teeth,  the  angle  of  face  of  small  gear  is  59°  11'. 
method  o?  ob*      ^nc  following  method  of  obtaining  the  whole  diam- 
DiametCTh o1?  ter  °^  ^evel  gears  is  sometimes  preferred  : 
Blanks.  From  k  lay  off  ;  upon  the  cone-pitch  line,  a  distance 

K  w,  equal  to  ten  times  the  working  depth  of  the 
teeth  =  10  D".  Now  add  TO  of  the  shortest  distance 
of  w  from  the  line  g  h,  which  is  the  perpendicular 
dotted  line  w  x,  to  the  outside  pitch  diameter  of  gear, 
and  the  sum  will  be  the  whole  diameter  of  gear.  In 
the  same  manner  TO  of  w  y,  added  to  the  outside  pitch 
diameter  of  pinion,  gives  the  whole  diameter  of  pinion. 
The  part  added  to  the  pitch  diameter  is  called  the 
diameter  increment. 

Part  II  gives  trigonometrical  methods  of  figuring 
bevel  gears  :  in  our  Formulas  in  Gearing  there  are 
trigonometrical  formulas  for  bevel  gears,  and  also 
tables  for  angles  and  sizes. 

^  somewhat  similar  construction  will   do   for  bevel 
arehnot &ears  Wflose  axes  are  not  at  right  angles.' 

fes  RigM  An"     In  Fig>  18  tne  axes  are  snown  at   O  B  and  O  D,  the 
angle  BOD  being  less  than  a  right  angle. 

1.  Parallel  to  O  B,  and  at  a  distance  from  it  equal 
to  the  radius  of  the  gear,  we  draw  the  lines  a  b  and  c  d. 


PROVIDENCE,    R.    I. 


39 


B 
Fiff. 18 

ANGLE  OF  AXES  LESS  THAN  90°. 


ANGLE  OF  AXES  MORE 
THAN  90°. 


INSIDE  BEVEL  GEAR 

AND  PINION  Fig.20 


BROWN    &   SHAKPE   MFG.    CO. 

2.  Parallel  to  O  D,  and  at  a  distance  from  it  equal 
to  the  radius  of  the  pinion,  we  draw  the  lines  e  f  and  g  h. 

3.  Now,  through  the  point  j  at  the  intersection  of 
c  d  and  g  h,   we   draw   a  line   perpendicular  to  O  B. 
This  line  k  j,  limited  by  a  b  and  c  d,   represents   the 
largest  pitch  diameter  of  the  gear. 

Through  j  we  draw  a  line  perpendicular  to  O  D. 
This  line  j  1,  limited  by  e  f  and  g  h,  represents  the 
largest  pitch  diameter  of  the  pinion. 

4.  Through  the  point  k  at  the  intersection  of  a  b 
with  k  j,  we  draw  a  line  to  O,  a  line  from  j  to  O,  and 
another  from  1,  at  the  intersection  j  1  and  e  f  to  O. 
These  lines  O  k,  O  j,  and  O  1,  represent  the  cone- 
pitch  lines,  as  in  Fig.  17. 

5.  Perpendicular  to  the  cone-pitch  lines  we  draw 
the  lines  u  v,  o  p,  and  q  r.     Upon  these  lines  we  lay 
off  the  addenda  and  working  depth  as  in  the  previous 
figure,  and  then  draw  lines  to  the  point  O  as  before. 

By  a  similar  construction  Figs.  19  and  20  can  be 
drawn. 


GEAR  CUTTER. 


41 


CHAPTER  X. 

BEVEL  GEARS. 

FORMS  AND  SIZES  OF  TEETH. 
CUTTING  TEETH. 

To  obtain  the  form  of  the  teeth  in  a  bevel  srear  we      Form    of 

bevel    gear 

do  not  lay  them  out  upon  a  pitch  circle,  as  we  do  in  a  teeth, 
spur  gear,  because  the  rolling  pitch  surface  of  a  bevel 
gear,  at  any  point,  is  of  a  longer  radius  of  curvature 
than  the  actual  radius  of  a  pitch  circle  that  passes 
through  that  point.  Thus  in  Fig.  21,  let  f  g  c  be  a 
cone  about  the  axis  O  A,  the  diameter  of  the  cone 
being  f  c,  and  its  radius  g  c.  Now  the  radius  of 
curvature  of  the  surface,  at  c,  is  evidently  longer  than 
g  c,  as  can  be  seen  in  the  other  view  at  C  ;  the  full 
line  shows  the  curvature  of  the  surface,  and  the  dotted 
line  shows  the  curvature  of  a  circle  of  the  radius  g  c. 
It  is  extremely  difficult  to  represent  the  exact  form  of 
bevel  gear  teeth  upon  a  flat  surface,  because  a  bevel 
gear  is  essentially  spherical  in  its  nature  ;  for  practical 
purposes  we  draw  a  line  c  A  perpendicular  to  O  c, 
letting  c  A  reach  the  centre  line  O  A,  and  take  c  A 
as  the  radius  of  a  circle  upon  which  to  lay  out  the 
teeth.  This  is  shown  at  c  11  m,  Fig.  22.  For  con- 
venience the  line  c  A  is  sometimes  called  the  back 
cone  radius. 

Let  us  take,  for  an  example,  a  bevel  gear  and  a    FigX|>mple: 
pinion  24  and  18  teeth,  5  pitch,  shafts  at  right  angles. 
To  obtain  the  forms  of  the  teeth   and   the  data  for 
cutting,  we  need  to  draw  a  section  of  only  a  half  of 
each  gear,  as  in  Fig.  22. 


42  BROWN    &    SHAKPE   MFG.    CO., 

1.  Draw   the    centre    lines  A  O  and  B  O,  then  the 
lines  g  h  and  c  d,  and  the   gear  blank  lines  as  des- 
cribed in  Chapter  IX.     Extend  the  lines  o'  p'  and  o  p 
until  they  meet  the  centre  lines  at  A'  B'  and  A  B. 

2.  With  the  radius  A  c  draw  the  arc  c  n  m,  which 
we  take  as  the  geometrical  pitch  circle  upon  which  to 
lay  out  the  teeth  at  the  large  end.     The  distance  A'  c' 
is  taken  as  the  radius  of  the  geometrical  pitch  circle 
at  the  small  end  ;  to  avoid  confusion  an  arc  of  this 
circle  is  drawn  at  G"  n'  m'  about  A. 

3.  For  the  pinion  we  have  the  radius  B  c  for  the 
geometrical  pitch  circle  at  the  large  end  and  B'  c'  for 
the    small   end :    the  distance    B'  c'  is    transferred   to 
B  c'". 

4.  Upon  the  arc  c  n  m  lay  off  spaces  equal  to  the 
tooth  thickness  at  the  large  pitch  circle,  which  in  our 
example  is  .314".     Draw  the  outlines  of  the  teeth  as 
in  previous  chapters  :  for  single  curve  teeth  we  draw  a 
semi-circle  upon  the  radius  A  c,  and  proceed  as  des- 
cribed in  chapter  III.     For  all  bevel  gears  that  are  to 
be  cut  with  a  rotary  disk  cutter,   or    a   common  gear 
cutter,  single  curve  teeth  are  chosen  ;  and  no  attempt 
should  be  made  to  cut  double  curve   teeth.     Double 
curve  teeth  can   be  drawn  by   the  directions  given  in 
chapters   VII    and   VIII.      We  now  have  the  form  of 
the  teeth  at  the  large  end  of  the  gear.     Repeat  this 
operation  with  the  radius   B  C  about  B,  and  we  have 
the  form  of  the  teeth  at  the  large  end  of  the  pinion. 

5.  The  tooth  parts  at  the  small  end  are  designated 
by  the  same  letters  as  at  the  large,  with  the  addition 
of  an  accent  mark  to  each  letter,  as  in  the  right  hand 
column,  Fig.  2*2,  the  clearance,  f,  however,  is  usually 
the  same  at  the  small  end    as    at   the  large,  for  con- 
venience in  cutting  the  teeth. 

Sizes  of  the      The  sizes  of  the  tooth  parts  at  the  small  end  are  in 

tooth  parts. 

the  same  proportion  to  those  at  the  large  end  as 
the  line  O  c'  is  to  O  c.  In  our  example  O  c'  is  '2", 
and  O  c  is  3" ;  dividing  O  c'  by  O  c  we  have  f ,  or 
.666,  as  the  ratio  of  the  sizes  at  the  small  end  to  those 


PROVIDENCE,  R.  J. 


43 


44  BROWN  &  SHARPE  MFG.  CO. 

at  the  large  :  t'  is  .209"  or  f  of  .314%  and  so  on.  If 
the  distance  n  m  is  equal  to  the  outer  tooth  thickness, 
t,  upon  the  arc  c  n  m,  the  lines  n  A  and  m  A  will  be  a 
distance  apart  equal  to  the  inner  tooth  thickness  t' 
upon  the  arc  c"  n'  m'.  The  addendum,  s',  and  the 
working  depth,  D'",  are  at  o'  c'  and  o'  p'. 

6.     Upon  the  arcs  c"  n'  m  and  c'"  we  draw  the  forms 
of  the  teeth  of  the  gear  and  pinion  at  the  inside. 

Kxampie  of  As  an  example  of  the  cutting  of  bevel  gears  with 
rotary  disk  cutters,  or  common  gear  cutters,  let  us 
take  a  pair  of  8  pitch,  12  and  24  teeth,  shown  in 
Fig.  23. 

Length  of  In  making  the  drawing  it  is  well  to  remember  that 
nothing  is  gained  by  having  the  face  F  E  longer  than 
five  times  the  thickness  of  the  teeth  at  the  large 
pitch  circle,  and  that  even  this  is  too  long  when  it  is 
more  than  a  third  of  the  apex  distance  O  c.  To  cut  a 
bevel  gear  with  a  rotary  cutter,  as  in  Fig.  24,  is  at 
best  but  a  compromise,  because  the  teeth  change  pitch 
from  end  to  end,  so  that  the  cutter,  being  of  the  right 
form  for  the  large  ends  of  the  teeth  can  not  be  right 
for  the  small  ends,  and  the  variation  is  too  great  when 
the  length  of  face  is  greater  than  a  third  of  the  apex 
distance  O  c,  Fig.  23.  In  the  example,  one-third  of 
the  apex  distance  is  Ty,  but  F  E  is  drawn  only  a 
half  inch,  which  even  though  rather  short,  has  changed 
the  pitch  from  8  at  the  outside  to  finer  than  1 1  at  the 
inside.  Frequently  the  teeth  have  to  be  rounded  over 
at  the  small  ends  by  filing ;  the  longer  the  teeth  the 
more  we  have  to  file.  If  there  is  any  doubt  about  the 
strength  of  the  teeth,  it  is  better  to  lengthen  at  the 
large  end,  and  make  the  pitch  coarser  rather  than  to 
lengthen  at  the  small  end. 

Data  for       These  data  are  needed  before  beginning  to  cut : 

1 .  The  pitch  and  the  numbers  of  the  teeth  the  same 
as  for  spur  gears. 

2.  The  data  for  the  cutter,  as  to  its  form  :  some- 
times two  cutters  are  needed  for  a  pair  of  bevel  gears. 

3.  The  whole  depth  of  the  tooth   spaces,   both  at 


PROVIDENCE,    E.    I. 


45 


—  .209" 
S/=  .133" 
D"=  .266' 
S'+/   =  .165" 
D-+/   =.298" 


Fig.  22. 

BEVEL  GEARS,   FORM  AND  SIZE  OF  TEETH, 


46  BKOWN    &    SHARPE  MFG.  CO. 

the  outside  and   inside  ends ;  D"  -f  f  at  the  outside, 
and  D'"  -|-  f  at  the  inside. 

4.  The  thickness  of  the  teeth  at  the  outside  and  at 
the  inside  ;  t  and  t'. 

5.  The  height  of  the  teeth  above  the  pitch  lines  at 
the  outside  and  inside  ;  s  and  s'. 

6.  The  cutting  angles,  or  the  angles  that  the  path 
of  the  cutter  makes  with  the  axes   of  the  gears.     In 
Fig.  23  the  cutting  angle  for  the  gear  c  D  is   A  Op, 
and  the  cutting  angle  for  the  pinion  is  B  O  o. 

Selection  of       The  form  of  the  teeth  in  one  of  these  gears  differs 
cutters. 

so  much  from  that  in  the  other  gear  that  two  cutters 

are  required.  In  determining  these  cutters  we  do  not 
have  to  develop  the  forms  of  the  gear  teeth  as  in 
Fig.  22  ;  we  need  merely  measure  the  lines  A  c  and 
B  c,  Fig.  23,  and  calculate  the  cutter  forms  as  if  these 
distances  were  the  radii  of  the  pitch  circles  of  the 
gears  to  be  cut.  Twice  the  length  A  c,  in  inches, 
multiplied  by  the  diametral  pitch,  equals  the  number 
of  teeth  for  which  to  select  a  cutter  tor  the  twenty- 
four-tooth  gear ;  this  number  is  about  54,  which  calls 
for  a  number  three  bevel  gear  cutter  in  accordance 
with  the  lists  of  gear  cutters,  pages  61  and  82.  Twice 
B  c,  multiplied  by  8,  equals  about  13,  which  indicates 
a  No.  8  bevel  gear  cutter  for  the  pinion.  This  method 
of  selecting  cutters  is  based  upon  the  idea  of  shaping 
the  teeth  as  nearly  right  as  practicable  at  the  large  end, 
and  then  filing  the  small  end  where  ihe  cutter  has  not 
rounded  them  over  enough. 

In  Fig.  25  the  tooth  L  has  been  cut  to  thickness  at 
both  the  outer  and  inner  pitch  lines,  but  it  must  still 
be  rounded  at  the  inner  end.  The  teeth  M  M  have 
been  filed.  In  thus  rounding  the  teeth  they  should  not 
be  filed  thinner  at  the  pitch  lines. 

There  are  several  things  that  affect  the  shape  of  the 
teeth,  so  that  the  choice  of  cutters  is  not  always  so 
simple  a  matter  as  the  taking  of  the  lines  A  c  and 
B  c  as  radii. 

In  cutting  a  bevel  gear,  in  the  ordinary  gear  cutting 


PROVIDENCE,    R.    I. 


47 


BEVEL  GEAR  DIAGRAM   FOR  DIMENSIONS. 


48 


BROWN    &   SIIARPE   MFG.    CO. 


machines,  the  finished  spaces  are  not  always  of  the 
same  form  as  the  cutter  might  be  expected  to  make, 
because  of  the  changes  in  the  positions  of  the  cutter 
and  of  the  gear  blank  in  order  to  cut  the  teeth  of  the 
right  thickness  at  both  ends.  The  cutter  must  of 
course  be  thin  enough  to  pass  through  the  small  end  of 
the  spaces,  so  that  the  large  end  has  to  be  cut  to  the 
right  width  by  adjusting  either  the  cutter  or  the  blank 
side  wise,  then  rotating  the  blank  and  cutting  twice 
around. 

widening       Thus,  in  Fig.  24,  a  gear  and  a  cutter  are  set  to  have 
a  space  widened  at  the  large  end  e',  and  the  last  chip 


to  be  cut  off  by  the  right  side  of   the  cutter,  the  cutter 
having  been  moved  to  the  left,  and  the  blank  rotated 
in  the  direction  of  the  arrow  :  in  a  Universal  Milling 
Machine  the  same  result  would  be  attained  by  moving 
the  blank  to  the  right  and  rotating   it   in  the  direction 
of  the  arrow.     It  may   be   well   to  remember  that  in 
setting  to  finish  the  side  of  a  tooth,  the  tooth  and  the 
cutter  are  first  separated  side  wise,    and   the  blank   is 
then  rotated  by  indexing  the  spindle  to  bring  the  large 
Teeth  nar-  enc*  °^  ^ie  tootn  UP    against  the   cutter.     This  tends 
at^'ace  Than   nofc  on^   to   cut  tae  sPaces  wider   at  the  large   pitch 
at  root.  circle,  but  also  to  cut  off  still  more   at  the  face  of  the 

tooth  ;  that  is,  the  teeth  may  be  cut  rather  thin  at  the 
face  and  left  rather  thick  at  the  root.  This  tendency 
is  greater  as  a  cutting  angle  B  O  o,  Fig.  23,  is  smaller, 
or  as  a  bevel  gear  approaches  a  spur  gear,  because 
when  the  cutting  angle  is  small  the  blank  must  be 
rotated  through  a  greater  arc  in  order  to  set  to  cut  the 
right  thickness  at  the  outer  pitch  circle.  This  can  be 
understood  by  Figs.  '26  and  27.  Fig.  26  is  a  radial- 
toothed  clutch,  which  for  our  present  purpose  can  be 
regarded  as  one  extreme  of  a  bevel  gear  in  which  the 
teeth  are  cut  square  with  the  axis  :  the  dotted  lines 
indicate  the  different  positions  of  the  cutter,  the  side 
of  a  tooth  being  finished  by  the  side  of  the  cutter  that 
is  on  the  centre  line.  In  setting  to  cut  these  teeth 
there  is  the  same  side  adjustment  and  rotation  of  the 


PROVIDENCE,    R.    I. 


SETTING  BEVEL  GEAR  CUTTER 
OUT  OF  CENTRE. 


50 


BROWN    &    SHARPE   MFG.    CO. 


spindle  as  in  a  bevel  gear,  but  there  is  no  tendency  to 
make  a  tooth  thinner  at  the  face  than  at  the  root.  On 
the  other  hand,  if  we  apply  these  same  adjustments  to 
a  spur  gear  and  cutter,  Fig.  27,  we  shall  cut  the  face 
F  much  thinner  without  materially  changing  the  thick- 
ness of  the  root  R. 


Fig.  26 

Almost  all  bevel  gears  are  between  the  two  extremes 
of  Figs.  26  and  27,  so  that  when  the  cutting  angle 
B  O  o,  Fig.  23,  is  smaller  than  about  30°,  this  change 
in  the  form  of  the  spaces  caused  by  the  rotation  of  the 
blank  may  be  so  great  as  to  necessitate  the  substitution 


FINISHED  GEAR. 


PROVIDENCE,    R.    I.  51 

of  a  cutter  that  is  narrower  at  e  e',  Fig.  24,  than  is 
called  for  by  the  way  of  figuring  that  we  have  just 
given  :  thus  in  our  own  gear  cutting  department  we 
might  cut  the  pinion  with  a  No.  6  cutter,  instead  of  a 
No.  8.  The  No.  6,  being  for  17  to  20  teeth,  cuts  the 
tooth  sides  with  a  longer  radius  of  curvature  than  the 
No.  8,  which  may  necessitate  considerable  filing  at  the 
small  ends  of  the  teeth  in  order  to  round  them  over 
enough.  Fig.  28  shows  the  same  gear  as  Fig.  25,  but 
in  this  case  the  teeth  have  all  been  filed  similar  to 
M  M,  Fig.  25. 

Different  workmen   prefer   different   ways   to  com-      Filing  the 


promise   in   the  cutting   of   a   bevel   gear.     When   a         i  end!he 


blank  is  rotated  in  adjusting  to  finish  the  large  end  of 
the  teeth  there  need  not  be  much  filing  of  the  small 
end,  if  the  cutter  is  right,  for  a  pitch  circle  of  the 
radius  B  c,  Fig.  23,  which  for  our  example  is  a  No.  8 
cutter,  but  the  tooth  faces  may  be  rather  thin  at  the 
large  ends.  This  compromise  is  preferred  by  nearly 
all  workmen,  because  it  does  not  require  much  filing 
of  the  teeth  :  it  is  the  same  as  is  in  our  catalogue  by 
which  we  fill  any  order  for  bevel  gear  cutters,  unless 
otherwise  specified.  This  means  that  we  should  send  selection  of 

cutter   when 

a  No.  8,  8-pitch  bevel  gear  cutter  in  reply  to  an  order  j^1}^1"6  to 

for  a  cutter  to  cut  the  12-tooth  pinion,  Fig.  23  ;    while 

in  our  own  gear  cutting  department  we  might  cut  the 

same  pinion  with  a  No.  6,  8-pitch  cutter,  because  we 

prefer  to  file   the  teeth  at  the  small  end  after  cutting 

them  to  the  right  thickness  at  the  faces  of  the  large 

end.     We  should  take  a  No.  6  instead  of  a  No.  8  only 

for  a  12-tooth  pinion  that  is  to  run  with  a  gear  two  or 

three  times  as  large.      We  generally  step  off  to  the 

next  cutter   for  pinions  fewer  than  twenty-five  teeth, 

when  the  number  for  the  teeth  has   a   fraction  nearly 

reaching  the  range   of  the  next  cutter  :  thus,  if  twice 

the  line  B  c   in   inches,    Fig.    23,   multiplied   by   the 

diametral  pitch,  equals  20.9,    we    should  use  a  No.  5 

cutter,    which   is   for  21    to  25   teeth   inclusive.      In 

filling  an  order  for  a  gear  cutter,  we  do  not  consider 


52 


BROWN    &    SHARPE   MFG.    CO. 


the  fraction  but  send  the  cutter  indicated  by  the  whole 
number. 

Later  on  we  will  refer  to  other  compromises  that  are 
made  in  the  cutting  of  bevel  gears. 

The  sizes  of  the  8-pitch  tooth  parts,  Fig.  23,  at  the 
large  end,  are  copied  from  the  table  of  spur  gear 
teeth,  pages  146  to  149. 

The  distance  Oc'  is  seven-tenths   of  the    apex  dis- 
tance Oc,  so  that  the   sizes  of  the  tooth   parts  at  the 

Form     of 

gear  cutting  small  end,  except  f,  are  seven-tenths  the  large.  The 
order  for  cutting  these  gears  goes  to  the  workmen  in 
this  form  : 

LARGE  GEAR. 


P  =  8 

N  =  24 

D" 

-f  f  =  .'270 

n 

D'" 

4- 

f 

= 

.195" 

t  =  .196 

n 

f 

= 

.137" 

s   =  .125 

s' 

= 

.087" 

Cutting 

Angle  = 

59°  10' 

SMALL  GEAR. 

N  =  12 

Cutting  Angle  =  22°  18' 

setting  the       Fig.  32  is  a  side  view  of    a  Gear  Cutting  Machine. 

machine.  A  beyel  gear  blank  A    ig  hdd  by  the    index  spindle  J> 

The  cutter  C  is  carried  by  the  cutter-slide  D.  The 
cutter-slide-carriage  E  can  be  set  to  the  cutting  angle, 
the  degrees  being  indicated  on  the  quadrant  F. 

Fig.  33  is  a  plan  of  the  machine  :  in  this  view  the 
cutter-slide-carriage,  in  order  to  show  the  details  a 
little  plainer,  is  not  set  to  an  angle. 

Before  beginning  to  cut  the  cutter  is  set  central  with 
the  index  spindle  and  the  dial  G  is  set  to  zero,  so 
that  we  can  adjust  the  cutter  to  any  required  distance 
out  of  centre,  in  either  direction.  Set  the  cutter-slide- 
carriage  E,  Fig.  32,  to  the  cutting  angle  of  the  gear, 
which  for  24-teeth  is  59°  10'  ;  the  quadrant  being 
divided  to  half -degrees,  we  estimate  that  10'  or  £  de- 


PROVIDENCE,  R.  I 


53 


gree  more  than  59o.  Mark  the  depth  of  the  cut  at  the 
outside,  as  in  Fig.  30  :  it  is  also  well  enough  to  mark 
the  depth  at 'he  inside  as  a  check.  The  thickness  of 
the  teeth  at  the  large  end  is  conveniently  deter- 
mined by  the  solid  gauge,  Fig.  29.  The  gear-tooth 


GEAR  TOOTH  GAUGE. 


DEPTH 
GAUGE. 


GEAR  TOOTH  CALIPER. 

JFiff.31 

vernier  caliper,  Fig.  31,  will  measure  the  thickness  of 
teeth  up  to  2  diametral  pitch.  In  the  absence  of  the 
vernier  caliper  we  can  file  a  gauge,  similar  to  Fig.  29, 
to  the  thickness  of  the  teeth  at  the  small  end. 

The  index  having  been  set  to  divide  to  the  right , 
number  we  cut  two  spaces  central  with  the  blank, being  flnished- 
leaving  a  tooth  between  that  is  a  little  too  thick,  as  in 
the  upper  part  of  Fig.  25.  If  the  gear  is  of  cast  iron, 
and  the  pitch  is  not  coarser  than  about  5  diametral, 
this  is  as  far  as  we  go  with  the  central  cuts,  and  we 
proceed  to  set  the  cutter  and  the  blank  to  finish  first 
one  side  of  the  teeth  and  then  the  other,  going  around 
only  twice.  The  tooth  has  to  be  cut  away  more  in 
proportion  from  the  large  than  from  the  small  end, 
which  is  the  reason  for  setting  the  cutter  out  of  centre, 
as  in  Fig.  24. 


54 


BROWN    &    SHARPE    MFG.    CO. 


Fly.  32. 


AUTOMATIC  GEAR   CUTTING    MACHINE. 


SIDE  ELEVATION. 


PROVIDENCE,    R.    I.  65 

It  is  important  to  remember  that  the  part  of  the 
cutter  that  is  finishing  one  side  of  a  tooth  at  the  pitch 
line  should  be  central  with  the  gear  blank,  in  order  to 
know  at  once  in  which  direction  to  set  the  cutter  out  of 
centre.  We  can  not  readily  tell  how  much  out  of 
centre  to  set  the  cutter  until  we  have  cut  and  tried, 
because  the  same  part  of  a  cutter  does  not  cut  to  the 
pitch  line  at  both  ends  of  a  tooth.  As  a  trial  distance 
out  of  centre  we  can  take  about  one-tenth  to  one- 
eighth  of  the  thickness  of  the  teeth  at  the  large  end. 
The  actual  distance  out  of  centre  for  the  12-tooth 
pinion  is  .021":  for  the  24-tooth  gear,  .030",  when 
using  cutters  listed  in  our  catalogue. 

After  a  little  practice  a  workman  can  set  his  cutter  c^et°j;8clS8°f 
the  trial  distance  out  of  centre,  and  take  his  first  cuts, 
without  any  central  cuts  at  all ;  but  it  is  safer  to  take 
central  cuts  like  the  upper  ones  in  Fig.  25.  The 
depth  of  cut  is  partly  controlled  by  the  index-spindle 
raising-dial-shaft  H,  Fig.  33,  which  determines  the 
height  of  the  index  spindle,  and  partly  by  the  position 
of  the  cutter  spindle.  We  now  set  the  cutter  out  of 
centre  the  trial  distance  by  means  of  the  cutter- spindle 
dial-shaft,  I,  Fig.  33.  The  trial  distance  can  be  about 
one-tenth  the  thickness  of  the  tooth  at  the  large  end 
in  a  12-tooth  pinion,  and  from  that  to  one-eighth  the 
thickness  in  a  24-tooth  gear  and  larger.  The  principle 
of  trimming  the  teeth  more  at  the  large  end  than  at 
the  small  is  illustrated  in  Fig.  24,  which  is  to  move 
the  cutter  away  from  the  tooth  to  be  trimmed,  and 
then  to  bring  the  tooth  up  against  the  cutter  by 
rotating  the  blank  in  the  direction  of  the  arrow.  Adjustments. 

The  rotative  adjustment  of  the  index  spindle  is 
accomplished  by  loosening  the  connection  between  the 
index  worm  and  the  index  drive,  and  turning  the  worm  : 
the  connection  is  then  fastened  again.  The  cutter  is 
now  set  the  same  distance  out  of  centre  in  the  other 
direction,  the  index  spindle  is  adjusted  to  trim  the 
other  side  of  the  tooth  until  one  end  is  down  nearly 
to  the  right  thickness.  If  now  the  thickness  of  the 


56  BROWX   &   SHARPE   MFG.    CO. 

small  end  is  in  the  same  proportion  to  the  large  end  as 
Oc'  is  to  Oc,  Fig.  23,  we  can  at  once  adjust  to  trim 
the  tooth  to  the  right  thickness.  But  if  we  find  that 
the  large  end  is  still  going  to  be  too  thick  when  the 
small  end  is  right,  the  out  of  centre  must  be  increased. 

It  is  well  to  remember  this  :  too  much  out  of  centre 
leaves  the  small  end  proportionally  too  thick,  and  too 
little  out  of  centre  leaves  the  small  end  too  thin. 

After  the  proper  distance  out  of  centre  has  been 
learned  the  teeth  can  be  finish-cut  by  going  around  out 
of  centre  first  on  one  side  and  then  on  the  other  with- 
out cutting  any  central  spaces  at  all.  The  cutter 
spindle  stops,  J  J,  can  now  be  set  to  control  the  out 
of  centre  of  the  cutter,  without  having  to  adjust  by 
the  dial  G.  If,  however,  a  cast  iron  gear  is  5-pitch 
or  coarser  it  is  usually  well  to  cut  central  spaces  first 
and  then  take  the  two  out-of -centre  cuts,  going  around 
three  times  in  all.  Steel  gears  should  be  cut  three 
times  around. 

Blanks  are  not  always  turned  nearly  enough  alike  to 
be  cut  without  a  different  setting  for  different  blanks. 
If  the  hubs  vary  in  length  the  position  of  the  cutter 
spindle  has  to  be  varied.  In  thus  varying,  the  same 
depth  of  cut  or  the  exact  D"  -\-  f  may  not  always  be 
reached.  A  slight  difference  in  the  depth  is  not  so 
objectionable  as  the  incorrect  tooth  thickness  that  it 
may  cause.  Hence,  it  is  well,  after  cutting  once 
around  and  finishing  one  side  of  the  teeth,  to  give 
careful  attention  to  the  rotative  adjustment  of  the 
index  spindle  so  as  to  cut  the  right  thickness. 

After  a  gear  is  cut,  and  before  the  teeth  are  filed,  it 
is  jiot  always  a  very  satisfactory-looking  piece  of  work. 
In  Fig.  25  the  tooth  L  is  as  the  cutter  left  it,  and  is 
ready  to  be  filed  to  the  sh  ipe  of  the  teeth  MM,  which 
have  been  filed.  Fig.  34  is  the  pair  of  gears  that  we 
have  been  cutting ;  the  teeth  of  the  12-tooth  pinion 
have  been  filed. 


PROVIDENCE,    R.    I. 


UJ 


X 

•o 


UJ 

O 
O 

< 

2 

c 


58  BROWN    &    SHARPE   MFG.    CO. 

approxima?       ^   second  approximation  in  cutting  with  a   rotary 

Hon.  cutter  is  to  widen  the  spaces  at  the  large  end  by  swing- 

ing either  the  index  spindle  or  the  cutter-slide-carriage, 
so  as  to  pass  the  cutter  through  on  an  angle  with  the 
blank  side-ways,  called  the  side-angle,  and  not  rotate 
the  blank  at  all  to  widen  the  spaces.  This  side-angle 
method  is  employed  in  our  No.  11  Automatic  Bevel 
Gear  Cutting  Machines :  it  is  available  in  the  manufac- 
ture of  bevel  gears  in  large  quantities,  because  with 
the  proper  relative  thickness  of  cutter,  the  tooth- 
thickness  comes  right  by  merely  adjusting  for  the 
side-angle ;  but  for  cutting  a  few  gears  it  is  not  much 
liked  by  workmen,  because,  in  adjusting  for  the  side- 
angle,  the  central  setting  of  the  cutter  is  usually  lost, 
and  has  to  be  found  by  guiding  into  the  central  slot 
already  cut.  If  the  side-angle  mechanism  pivots  about 
a  line  that  passes  very  near  the  small  end  of  the  tooth 
to  be  cut,  the  central  setting  of  the  cutter  may  not 
be  lost.  In  widening  the  spaces  at  the  large  end, 
the  teeth  are  narrowed  practically  the  same  amount  at 
the  root  as  at  the  face,  so  that  this  side-angle  method 
requires  a  wider  cutter  at  e  e',  Fig.  24,  than  the  first, 
or  rotative  method.  The  amount  of  filing  required 
to  correct  the  form  of  the  teeth  at  the  small  end  is 
about  the  same  as  in  the  first  method. 

A  third  ap-  A  third  approximate  method  consists  in  cutting 
the  teeth  right  at  the  large  end  by  going  around  at 
least  twice,  and  then  to  trim  the  teeth  at  the  small  end 
and  toward  the  large  with  another  cutter,  going  around 
at  least  four  times  in  all.  This  method  requires  skill 
and  is  necessarily  a  little  slow,  but  it  contains  possi- 
bilities for  considerable  accuracy. 

A  fourth  ap-  A  f ourth  method  is  to  have  a  cutter  fully  as  thick  as 
the  spaces  at  the  small  end,  cut  rather  deeper  than 
the  regular  depth  at  the  large  end,  and  go  only  once 
around.  This  is  a  quick  method  but  more  inaccurate 
than  the  three  preceding  :  it  is  available  in  the  manu- 
facture of  large  numbers  of  gears  when  the  tooth-face 


PROVIDENCE,    R.    I. 


59 


Fig.  34 


FINISHED  GEAR  AND   PINION 


60  BHOWN   &   SIIARPE   MFG.    CO. 

is  short  compared  with  the  apex  distance.  It  is  little 
liked,  and  seldom  employed  in  cutting  a  few  gears  :  it 
may  require  some  experimenting  to  determine  the  form 
of  cutter.  Sometimes  the  teeth  are  not  cut  to  the 
regular  depth  at  the  small  end  in  order  to  have  them 
thick  enough,  which  may  necessitate  reducing  the 
addendum  of  the  teeth,  s',  at  the  small  end  by  turning 
the  blank  down.  This  method  is  extensively  employed 
by  chuck  manufacturers. 

A  machine  that  cuts  bevel  gears  with  a  reciprocating 
motion  and  using  a  tool  similar  to  a  planer  tool  is 
called  a  Gear  Planer  and  the  gears  so  cut  are  said  to 
be  planed. 

Planing  of      ^ne  f°rm  of  Gear  Planer  is  that  in  which  the  prin- 

bevei  gears.    G^e  embodied  is  theoretically  correct ;  this  machine 

originates  the  tooth  curves  without  a  former.    Another 

form  of  the  same  class  of  machines  is  that  in  which  the 

tool  is  guided  by  a  former. 

Usually  the  time  consumed  in  planing  a  bevel  gear 
is  greater  than  the  time  necessary  to  cut  the  same  gear 
with  a  rotary  cutter,  thus  proportionately  increasing 
the  cost. 

Pitches  coarser  than  4  are  more  correct  and  some- 
times less  expensive  when  planed  ;  it  is  hardly  prac- 
ticable, and  certainly  not  economical,  to  cut  a  bevel 
gear  as  coarse  as3P.  with  a  rotary  cutter.  In  gears  as 
fine  as  16P.  planing  affords  no  practical  gain  in  quality. 

While  planing  is  theoretically  correct,  yet  the  wear- 
ing of  the  tool  may  cause  more  variation  in  the  thick- 
ness of  the  teeth  than  the  wearing  of  a  rotary  cutter, 
and  even  a  planed  gear  is  sometimes  improved  by  filing. 
Mounting  of  ^  gears  are  not  correctly  mounted  in  the  place  where 
they  are  to  run,  they  might  as  well  not  be  planed.  In 
fact,  after  taking  pains  in  the  cutting  of  any  gear, 
when  we  come  to  the  mounting  of  it  we  should  keep 
right  on  taking  pains. 

Angles  and      The  method  of  obtaining  the  sizes  and  angles  per- 
0        6  taining  to  bevel  gears  by  measuring  a  drawing  is  quite 
convenient,    and   with   care   is   fairly   accurate.      Its 


PROVIDENCE,    R. 


61 


accuracy  depends,  of  course,  upon  the  careful  measur- 
ing of  a  good  drawing.  We  may  say,  in  general,  that 
in  measuring  a  diagram,  while  we  can  hardly  obtain 
data  mathematically  exact,  we  are  not  likely  to  make 
wild  mistakes.  Some  years  ago  we  depended  almost 
entirely  upon  measuring,  but  since  the  publication  of 
this  "Treatise"  and  our  "  Formulas  in  Gearing  "  we 
calculate  the  data  without  any  measuring  of  a  drawing. 
In  the  "  Formulas  in  Gearing"  there  are  also  tables 
pertaining  to  bevel  gears. 

Several  of  the  cuts  and  some  of  the  matter  in  this 
chapter  are  taken  from  an  article  by  O.  J.  Beale,  in 
the  "American  Machinist,"  June  20,  1895. 


CUTTERS  FOR 
MITRE  AND  BEVEL  GEARS. 


Diametral  Pitch. 

Diameter  of  Cutter. 

Hole  in  Cutter. 

4 

3  1-2* 

1   1-4" 

5 

3  1-2 

1   1-4 

6 

3  1-2 

1  1-4 

7 

3  1-2 

1  1-4 

8 

3  1-4 

1  1-4 

10 

3  1-4 

7-8 

12 

3 

7-8 

14 

3 

7-8 

16 

2  3-4 

7-8 

20 

2  1-2 

7-8 

24 

2  1-4 

7-8 

When  each  gear  of  a  pair  of  bevel  gears  is  of  the 
same  size  and  the  gears  connect  shafts  that  are  at  right 
angles,  the  gears  are  called  "Mitre  Gears"  and  one 
cutter  will  answer  for  both. 


BROWN    &    SHARPE    MFG.    CO. 


WORM   WHEEL. 


Number  of  Teeth,  54. 
Throat  Diameter,  44.59". 


Circular  Pitch,  2£. 
Outside  Diameter,  46''. 


C3 


CHAPTER  XI. 
WORM  WHEELS— SIZING  BLANKS  OF  32  TEETH  AND  MORE, 


A  WOKM  is  a  screw  made  to  mesh,  with  the  teeth  of  Worm. 
a  wheel  called  a  worm-wheel.  As  implied  at  the  end  of 
Chapter  IV.,  a  section  of  a  worm  through  its  axis  is,  in 
outline,  the  same  as  a  rack  of  corresponding  pitch. 
This  outline  can  be  made  either  to  mesh  with  single  or 
double  curve  gear  teeth ;  but  worms  are  usually  made 
for  single  curve,  because,  the  sides  of  involute  rack 
teeth  being  straight  (see  Chapter  IV.),  the  tool  for 
cutting  worm-thread  is  more  easily  made.  The  thread- 
tool  is  not  usually  rounded  for  giving  fillets  at  bottom 
of  worm-thread. 

The  axis  of  a  worm  is  usually  at  right  jingles  to  the 
axis  of  a  worm  wheel:  no  other  angle  of  axis  is  treated 
of  in  this  book. 

The  rules  for  circular  pitch  apply  in  the  size  of  tooth 
parts  and  diameter  of  pitch-circle  of  worm-wheel. 

The  pitch  of  a  worm  or  screw  is  sometimes  given  in  Pitch  of  worm 
a  way  different  from  the  pitch  of  a  gear,  viz. :  in  num- 
ber of  threads  to  one  inch  of  the  length  of  the  worm  or 
screw.  Thus,  to  say  a  worm  is  2  pitch  may  mean  2 
threads  to  the  inch,  or  that  the  worm  makes  two  turns 
to  advance  the  thread  one  inch.  But  a  worm  may  be 
double- threaded,  triple-threaded,  and  so  on;  hence 
to  avoid  misunderstanding,  it  is  better  always  to  call 
the  advance  of  the  worm  thread  the  lead.  Thus,  a Worem-Threfad. 
worm-thread  that  advances  one  inch  in  one  turn  we 
call  one-inch  lead  in  one  turn.  A  single-thread  worm 
4  turns  to  1"  is  £"  lead.  We  apply  the  term  pitch,  that  is 
the  circular  pitch,  to  the  actual  distance  between  the 
threads  or  teeth,  as  in  previous  chapters.  In  single- 
thread  worms  the  lead  and  the  pitch  are  alike.  If  we 
have  to  make  a  worm  and  wheel  so  many  threads  to 


BROWN    &    SHAKPE    MFG.    CO. 


FIG.  35 -WORM  AND  WORM-WHEEL 

THE  THREAD  OF  WORM  is  LEFT-HANDED;  WORM  is  SINGLE-THREADED. 


PEOVIDENCE,    K.    I. 


G5 


QQ  BPvO\\N    &    SHARPE    MFG.    CO. 

one  inch,  we  first  divide  1"  l)y  the  number  of  threads  to 
one  inch)  and  the  quotient  is  the  circular  pitch.    Hence, 

Linear  pitch,  the  wheel  in  Fig.  36  is  y  circular  pitch.  Linear  pitch 
expresses  exactly  what  is  meant  by  circular  pitch. 
Linear  pitch  has  the  advantage  of  being  an  exact  use 
of  language  when  applied  to  worms  and  racks.  The 
number  of  threads  to  one  inch  linear,  is  the  reciprocal 
of  the  linear  pitch. 

Multiply  3.1416  by  the  number  of  threads  to  one 
inch,  and  the  product  will  be  the  diametral  pitch  of  the 
worm-wheel.  Thus,  we  should  say  of  a  double-threaded 
worm  advancing  1"  in  l£  turns  that: 

Drawing  of     Lead=f"  or  .75".    Linear  pitch  or  P'=f"  or  .375". 
worm^wheeL        Diametral  pitch  or  P = 8. 377.    See  table  of  tooth  parts. 
To  make  drawing  of    worm   and   wheel  we  obtain 
data  as  in  circular  pitch. 

1.  Draw  center  line  A  O  and  upon  it  space  off  the 
distance  a  b  equal  to  the  diameter  of  pitch-circle. 

2.  On  each  side  of  these  two  points  lay  off  the  dis- 
tance s,  or  the  usual  addendum =j  ',  as  b  c  and  b  d. 

3.  From   c  lay  off  the   distance  c  O  equal  to   the 
radius  of  the  worm.     The  diameter  of  a  worm  is  gen- 
erally four  or  five  times  the  circular  pitch. 

4.  Lay  off  the  distances  c  g  and  d  e  each  equal  to/, 
or  the  usual  clearance  at  bottom  of  tooth  space. 

5.  Through  c  and  e  draw  circles  about  O.     These 
represent  the  whole  diameter  of  worm  and  the  diam- 
eter at  bottom  of  worm- thread. 

6.  Draw  h  O  and  i  O  at  an  angle  of  30°  to  45°  with 
A  O.     These  lines  give  width  of  face  of  worm-wheel. 

7.  Through  g  and  d  draw  arcs  about  O,  ending  in 
h  O  and  i  O. 

This  operation  repeated  at  a  completes  the  outline 
of  worm-wheel.  For  32  teeth  and  more,  the  addendum 
diameter,  or  D,  should  be  taken  at  the  throat  or 
smallest  diameter  of  wheel,  as  in  Fig.  36.  Measure 
sketch  for  whole  diameter  of  wheel-blank. 

Teeth    of     The  foregoing  instructions  and  sketch  are  for  cases 

Whedwithf Hob"  where  the  teeth  of  the  wheels  are  finished  with  a  hob. 

Hob.  A.  HOB  is  shown   in   Fig.    37,  being  a  steel  piece 


PROVIDENCE,  R.  I.  67 

threaded  with  a  tool  of  the  same  angle  as  the  tool  that 
threads  the  worm,  the  end  of  the  tool  being  .335  of 
the  linear  pitch ;  the  hob  is  then  grooved  to  make  teeth 
for  cutting,  and  hardened. 

The  whole  diameter  of  hob  should  be  at  least  2/,  Proportionsof 
or  twice  the  clearance  larger  than  the  worm.  In  our 
relieved  hobs  the  diameter  is  made  about  .005"  to  .010" 
larger  to  allow  for  wear.  The  outer  corners  of  hob-thread 
can  be  rounded  down  as  far  as  the  clearance  distance. 
The  width  at  top  of  the  hob-thread  before  rounding 
should  be  .31  of  the  linear,  or  circular  pitch  =.3 IF. 
The  whole  depth  of  thread  is  thus  the  ordinary  work- 
ing depth  plus  the  clearance=D"+/*.  The  diameter 
at  bottom  of  hob-thread  should  be  2/+.005"  to  .010" 
larger  than  the  diameter  at  bottom  of  worm-thread. 


Fig.  37.— HOB. 

For  thread-tool  and  worm-thread  see  end  of  Chapter 
IV. 

In  the  absence  of  a  special  worm  gear  cutting  rna"t 
chine,  the  teeth  of  the  wheel  are  first  cut  as  nearly  to  the 
finished  form  as  practicable;  the  hob  and  worm-wheel 
are  mounted  upon  shafts  and  hob  placed  in  mesh,  it  is 
then  rotated  and  dropped  deeper  into  the  wheel  until  the 
teeth  are  finished.  The  hob  generally  drives  the  worm- 
wheel  during  this  operation.  The  Universal  Milling  Ma-  universal 
chine  is  convenient  for  doing  this  work  :  with  it  the  dis-  chine1 'use? 8i 

Hobbing. 


68 


BROWN    &    SiJARPE   MFG.   CO. 


Fig.  38. 


PROVIDENCE,    K.    I. 


GO 


Fig.  39. 


70  BROWN    &    SHARPE    MFG.    CO. 

tance  between  axes  of  worm  and  wheel  can  be  noted.  In 
making  wheels  in  quantities  it  is  belter  to  have  a  ma- 
chine in  which  the  work  spindle  is  driven  by  gearing, 
so  that  the  hob  can  cut  the  teeth  from  the  solid  with- 
whyawheei ou^  gasning-  The  object  of  nobbing  a  wheel  is  to  get 

isHobbed.       more  bearing  surface  of  the  teeth  upon  worm-thread. 

The  worm-wheels,  Figs.  35  and  43,  were  bobbed, 
worm-wheel      If  we  make  the  diameter  of  a  worm-wheel  blank,  that 

Less   than    so  is  to  have  less  than  30   teeth,  by   the  common   rules 

for  sizing  blanks,  and  finish  the  teeth  with  a  hob,  we 

shall  find  the  flanks  of  teeth  near  the  bottom  to  be  un- 

interference  dercut  or  hollowing.     This  is  caused  by  the  interfer- 

Fiank.  ence  spoken  of  in  Chapter  VI.     Thirty  teeth  was  there 

given  as  a  limit,  which  will  be  right  when  teeth  are 
made  to  circle  arcs.  With  pressure  angle  14  J°,  and 
rack-teeth  with  usual  addendum,  this  interference  of 
rack-teeth  with  flanks  of  gear-teeth  begins  at  31  teeth 
(31 A  geometrically),  and  interferences  with  nearly  the 
whole  flank  in  wheel  of  12  teeth. 

Fig.38.  In  Fig  38  the  blank  for  worm-wheel  of  12  teeth  was 

sized  by  the  same  rule  as  given  for  Fig.  36.  The  wheel 
and  worm  are  sectioned  to  show  shape  of  teeth  at  the 
mid-plane  of  wheel.  The  flanks  of  teeth  are  undercut 
by  the  hob.  The  worm-thread  does  not  have  a  good 
bearing  on  flanks  inside  of  A,  the  bearing  being  that  of 
a  corner  against  a  surface. 

Fig.  39.  ln  Fig  39  the  blank  for  wheel  was  sized  so  that  pitch- 

circle  comes  midway  between  outermost  part  of  teeth 
•    and  innermost  point  obtained  by  worm  thread. 

This  rule  for  sizing  worm-wheel  blanks  has  been  in 
use  to  some  extent.  The  hob  has  cut  away  flanks  of 
teeth  still  more  than  in  Fig.  38.  The  pitch  circle  in 
Fig.  39  is  the  same  diameter  as  the  pitch-circle  in  Fig. 
38.  The  same  hob  was  used  for  both  wheels.  The 
flanks  in  this  wheel  are  so  much  undercut  as  to  mate- 
rially lessen  the  bearing  surface  of  teeth  and  worm- 
thread. 

interference  In  Chapter  VI.  the  interference  of  teeth  in  high- 
numbered  gears  and  racks  with  flanks  of  12  teeth  was 
remedied  by  rounding  off  the  addenda.  Although  it 
would  be  more  systematic  to  round  off  the  threads  oJ 
a  worm,  making  them,  like  rack-teeth,  to  mesh  with 


PROVIDENCE,    R.   I.  71 

interchangeable  gears,  yet  this  has  not  generally  been 
done,  because  it  is  easier  to  make  a  worm-thread  tool 
with  straight  sides. 

Instead  of  cutting  away  the  addenda  of  worm- 
thread,  we  can  avoid  the  interference  with  flanks  of 
wheels  having  less  than  30  teeth  by  making  wheel 
blanks  larger. 

The  flanks  of  wheel  in  Fig.  40  are  not  undercut,  be-  Fig.  40. 
cause  the  diameter  of  wheel  is  so  large  that  there  is 
hardly  any  tooth  inside  the  pitch-circle.  The 
pitch-circle  in  Fig.  40  is  the  same  size  as  pitch- 
circles  in  Figs.  38  and  39.  This  wheel  was  sized 
by  the  f  ollowing  rule  :  Multiply  the  pitch  diameter  of  Diameter  at 

•/         i       IT         rtorr  T       -in    1        ,1  Throat  to  Avoid 

the  wheel  by  .9o7,  and  add  to  the  product  four  times  interference, 
the  addendum  (4  s) ;  the  sum  will  be  the  diameter  for 
the  blank  at  the  throat  or  small   part.     To  get  the 
whole  diameter,  make  a  sketch  with  diameter  of  throat 
to  the  foregoing  rule  and  measure  the  sketch. 

It  is  impractical  to  hob  a  wheel  of  12  to  about  16  or 
18  teeth  when  blank  is  sized  by  this  rule,  unless  the 
wheel  is  driven  by  independent  mechanism  and  not  by 
the  hob.  The  diameter  across  the  outermost  parts  of 
teeth,  as  at  A  B,  is  considerably  less  than  the  largest 
diameter  of  wheel  before  it  was  hobbed. 

In  general  it  is  well  to  size  all  blanks,  as  by  page  66 
and  Figs.  36  and  38,  when  the  wheels  are  to  be  hobbed  ; 
of  course  the  cutter  should  be  thin  enough  to  leave 
stock  for  finishing.  The  spaces  can  be  cut  the  full 
depth,  the  cutter  being  dropped  in. 

When  worm-wheels  are  not  hobbed  it  is  better  to 
turn  blanks  like  a  spur-wheel.  Little  is  gained  by 
having  wheels  curved  to  fit  worm  unless  teeth  are  fin- 
ished with  a  hob.  The  teeth  can  be  cut  in  a  straight 
path  diagonally  across  face  of  blank,  to  fit  angle  of 
worm-thread,  as  in  Figs.  41  and  44. 

In  setting  a  cutter  to  gash  a  worm  wheel,  Figs. 42  and 
45,  the  angle  is  measured  from  the  axis  of  the  worm- MacWues- 
wheel  and   the  angle  of  the  worm  thread  is,  in  conse- 
quence, measured  from  the  perpendicular  to  the  axis 
of  the  worm.     See  Chapter  V,  part  II. 


BEO\VN    &    FHARPE    MFG.    CO. 


ig.  40. 


PROVIDENCE,   R.   I.  73 

Some  mechanics  prefer  to  make  dividing  wheels  in 
two  parts,  joined  in  a  plane  perpendicular  to  axis,  hob 
teeth ,  then  turn  one  part  round  upon  the  other,  match 
teeth  and  fasten  parts  together  in  the  new  position, 
and  hob  again  with  a  view  to  eliminate  errors.  With 
an  accurate  cutting  engine  we  have  found  wheels  like 
Figs.  42  and  45,  not  bobbed,  every  way  satisfactory. 
As  to  the  different  wheels,  Figs.  43,  44  and  45,  when  anFdlggres43' ** 
worm  is  in  right  position  at  the  start,  the  life-time 
of  Fig.  43,  under  heavy  and  continuous  work,  will  be 
the  longest. 

Fig.  44  can  be  run  in  mesh  with  a  gear  or  a  rack  as 
well  as  with  a  worm  when  made  within  the  angular 
limits  commonly  required.  Strictly,  neither  two  gears 
made  in  this  way,  nor  a  gear  and  a  rack  would  be 
mathematically  exact,  as  they  might  bear  at  the  sides 
of  the  gear  or  at  the  ends  of  the  teeth  only  and  not  in 
the  middle.  At  the  start  the  contact  of  teeth  in  this 
wheel  upon  worm-thread  is  in  points  only;  yet  such 
wheels  have  been  many  years  successfully  used  in  ele- 
vators. 

Fig.  45  is  a   neat-looking  wheel.     In  gear  cutting 

engines  where  the  workman  has  occasion  to  turn  the 
work  spindle  by  hand,  it  is  not  so  rough  to  take  hold 
of  as  Figs.  43  and  44.  The  teeth  are  less  liable  to  in- 
jury than  the  teeth  of  Figs.  43  and 44. 


BROWN    &    SHARPE    MFG.    CO. 


Fig.  41. 

Worm-wheel  with  teeth  cut  in  a  straight  path  diagonally  across  face. 
Worm  is  double-threaded. 


PROVIDENCE,   R.   I. 


75 


Fig.  42. 

Worm  and  Worm- Wheel,  for  Gear-cutting  Engine. 


76 


BROWN    &    SHARrE    MFG.    CO. 


43. 


Fig.  44. 


Fig.  45. 


PROVIDENCE,   R.    I.  77 

Some  designers  prefer  to  take  off  the  outermost  part 
of  teeth  in  wheels  (Figs.  35  and  43),  as  shown  in  these 
two  figures,  and  not  leave  them  sharp,  as  in  Figs.  36 
and  39. 

We  do  not  know  that  this  serves  any  purpose  except 
a  matter  of  looks. 

In  ordering  worms  and  worm  wheels  the  centre  dis- 
tances should  be  given. 

If  there  can  be  any  limit  allowed  in  the  centre  distance 
it  should  be  so  stated. 

For  instance,  the  distance  from  the  centre  of  a  worm 
to  the  centre  of  a  worm  wheel  might  be  calculated  at 
6"  but  5  31-32"  or  6  1-32"  might  answer. 

By  stating  all  the  limits  that  can  be  allowed,  there 
may  be  a  saving  in  the  cost  of  work  because  time  need 
not  be  wasted  in  trying  to  make  work  within  narrower 
limits  than  are  necessary. 


HOBS  WITH  RELIEVED  TEETH. 

We  make  hobs  of  any  size  with  the  teeth  relieved  the 
same  as  our  gear  cutters.  The  teeth  can  be  ground  on 
their  faces  without  changing  their  form.  The  hobs  are 
made  with  a  precision  screw  so  that  the  pitch  of  the 
thread  is  accurate  before  hardening. 


78 


BROWN   &   SHARPE    M^Q.    CO. 


GASHING  TEETH  OF  HOB. 
10  Inches  Outside  Diameter. 


\ 


79 


CHAPTER  XII. 


SIZING  GEARS  WHEN  THE  DISTANCE  BETWEEN  CENTRES  AND  THE 
RATIOS  OF  SPEEDS  ARE  FIXED-GENERAL  REMARKS— WIDTH 
OF  FACE  OF  SPUR  GEARS— SPEED  OF  GEAR  COTTERS— TABLE 
OF  TOOTH  PARTS. 


Let  us  suppose  that  we  have  two  shafts  14 "  apart, 
center  to  center,  and  wish  to  connect  them  by  gears  so  tanc?and  Ratio 
that  they  will  have  speed  ratio  6  to  1.  We  add  the  6  fixed- 
and  1  together,  and  divide  14"  by  the  sum  and  get  2" 
for  a  quotient;  this  2",  multiplied  by  6,  gives  us  the 
radius  of  pitch  circle  of  large  wheel  =  12".  In  the  same 
manner  we  get  2"  as  radius  of  pitch  circle  of  small  wheel. 
Doubling  the  radius  of  each  gear,  we  obtain  24"  and  4" 
as  the  pitch  diameters  of  the  two  wheels.  The  two  num- 
bers that  form  a  ratio  are  called  the  terms  of  the  ratio. 
We  have  now  the  rule  for  obtaining  pitch-circle  diame- 
ter of  two  wheels  of  a  given  ratio  to  connect  shafts  a 
given  distance  apart : 

Divide  the  center  distance  by  the  sum  of  the  terms  Of 
the  ratio;  find  the  product  of  twice  the  quotient  by  each 
term  separately,  and  the  two  products  will  be  the  pitch 
diameters  of  the  two  wheels. 

It  is  well  to  give  special  attention  to  learning  the 
rules  for  sizing  blanks  and  teeth ;  these  are  much 
oftener  needed  than  the  method  of  forming  tooth  out- 
lines. 


80  BROWN    &    SHAEPK   MFG.    CO. 

A  blank  1£"  diameter  is  to  have  16  teeth :  what  will 
the  pitch  be?  What  will  be  the  diameter  of  the  pitch 
circle?  See  Chapter  V. 

A  good  practice  will  be  to  compute  a  table  of  tooth 
parts.  The  work  can  be  compared  with  the  tables 
pages  146-149. 

In  computing  it  is  well  to  take  7t  to  more  than  four 
places,  7t  to  nine  places  =  3.141592653.  ^  to  nine 
places  =  .318309886. 

There  is  no  such  thing  as  pure  rolling  contact  in 
teeth  of  wheels ;  they  always  rub,  and,  in  time,  will 
wear  themselves  out  of  shape  and  may  become  noisy. 

Bevel  gears,  when  correctly  formed,  run  smoother 
than  spur  gears  of  same  diameter  and  pitch,  because 
the  teeth  continue  in  contact  longer  than  the  teeth  of 
spur  gears.  For  this  reason  annular  gears  run  smoother 
than  either  bevel  or  spur  gears. 

Sometimes  gears  have  to  be  cut  a  little  deeper  than 
designed,  in  order  to  run  easily  on  their  shafts.  If 
any  departure  is  made  in  ratio  of  pitch  diameters  it  is 
better  to  have  the  driving  gear  the  larger,  that  is,  cut 
the  follower  smaller.  For  wheels  coarser  than  eisrht 

o 

diametral  pitch  (8  P),  it  is  generally  better  to  cut  twice 
around,  when  accurate  work  is  wanted,  also  for  large 
wheels,  as  the  expansion  of  parts  from  heat  often  causes 
inaccurate  work  when  cut  but  once  around.  There  is 
not  so  much  trouble  from  heat  in  plain  or  web  gears  as 
in  arm  gears. 

(^lrt?acifpur     The  width  of  face  of  cast-iron  gears  can,  for  general 
use,  be  made  2£  times  the  linear  pitch. 

In  small  gears  or  pinions  this  width  is  often  exceeded. 
Speed  of  Gear  The  speed  of  gear  cutters  is  subject  to  so  many  con- 
ditions that  definite  rules  cannot  be  given.  We  append 
a  table  of  average  speeds.  A  coarse  pitch  cutter  for 
pinion,  12  teeth,  would  usually  be  fed  slower  than  a 
cutter  for  a  large  gear  of  same  pitch. 


PROVIDENCE,  R.  I. 
TABLE  OF  AVERAGE  SPEEDS  FOR  GEAR-CUTTERS. 


81 


A 

to 

bflfl 

FH 

i-i      • 

d 

31 

s| 

1—  1  +J 

r3   3 

•8 

11 

4l 

gg    d 

Si 

c  ^^J 

S-S  fl' 

^  d  2rt- 

P 

JS 

C     OJ    4j 

Jiti 

1g5 

ffi^fl 

111 

||  || 

5 

. 

1 

i^ 

H'= 

^fi'^A 

^ 

2 

5      in. 

24 

18 

.025  in. 

.011  in. 

.  60  in. 

.  20  in. 

24 

4t,  " 

30 

24 

.028   « 

.013    " 

.84   « 

.31    " 

3 

36 

28 

.031   f' 

.015    " 

1.12    « 

.42   " 

4 

3|T  " 

42 

32 

.034   " 

.017    " 

1.43   " 

.54   " 

5 

31    " 

50 

40 

.037   " 

.019  ." 

1.85    " 

.76   « 

6 

2J-4-  " 

75 

55 

.030   " 

.016    " 

2.25    " 

.88    " 

7 

2_9    " 

85 

65 

.032   " 

.018    " 

2.72    " 

1.17   " 

8 

21     " 

95 

75 

.034   " 

.020    " 

3.23    " 

1.50   " 

10 

21     " 

125 

90 

.026   « 

.014    « 

3.25   « 

1.26   " 

12 

2       " 

135 

100 

.027   " 

.017    " 

3.64   " 

1.70   " 

20 

1-J     " 

145 

115 

.029   " 

.021    " 

4.20   " 

2.41    "' 

32 

1  ii       u 

160 

135 

.031   <; 

.025   " 

4.96   " 

3.37   " 

In  brass  the  speed  of  gear-cutters  can  be  twice  as  Bra£se  e  d  iu 
fast  as  in  cast  iron.  Clock-makers  and  those  making  a 
specialty  of  brass  gears  exceed  this  rate  even.  A  12  P 
cutter  has  been  run  1,200  (twelve  hundred)  turns  a 
minute  in  bronze.  A.  32  P  cutter  has  been  run  7,000 
(seven  thousand)  turns  a  minute  in  soft  brass. 

In  cutting  5  P  cast-iron  gears,  75  teeth,  a  No.  1,  6  Pfrom?Scticl.8 
cutter  was  run  136  (one  hundred  and  thirty-six)  turns 
a  minute,  roughing  the  spaces  out  the  full  5  P  depth  ; 
the  teeth  were  then  finished  with  a  5  P  cutter,  running 
208  (two  hundred  and  eight)  turns  a  minute,  feeding 
by  hand.  The  cutter  stood  well,  but,  of  course,  the 
cast  iron  was  quite  soft.  A  4  P  cutter  has  finished 
teeth  at  one  cut,  in  cast-iron  gears,  86  teeth,  running  48 
(forty-eight)  turns  a  minute  and  feeding  Ty  at  one 
turn,  or  3  in.  in  a  minute. 

Hence,  while  it  is  generally  safe  to  run  cutters  as  in 
the  table,  yet  when  many  gears  are  to  be  cut  it  is  well  to 
see  if  cutters  will  stand  a  higher  speed  and  more  feed. 

In  gears  coarser  than  3  P  it  is  more  economical  to 
cut  first  the  full  depth  with  a  stocking  cutter  and  then 
finish  with  a  gear  cutter.  This  stocking  cutter  is  made 


82  SHOWN   &    SHARPE   MFG.    CO. 

on  the  principle  of  a  circular  splitting  saw  for  wood. 
The  teeth,  however,  are  not  set ;  but  side  relief  is  ob- 
tained by  making  sides  of  cutter  blank  hollowing.  The 
shape  of  stocking  cutter  can  be  same  as  bottom  of 
spaces  in  a  12-tooth  gear,  and  the  thickness  of  cutter 
can  be  J  of  the  circular  pitch,  see  page  40. 

Keep  cutters  The  matter  of  keeping  cutters  sharp  is  so  important 
that  it  has  sometimes  been  found  best  to  have  the  work- 
man grind  them  at  stated  times,  and  not  wait  until  he 
can  see  that  the  cutters  are  dull.  Thus,  have  him 
grind  every  two  hours  or  after  cutting  a  stated  number 
of  gears.  Cutters  of  the  style  that  can  be  ground 
upon  their  tooth  faces  without  changing  form  are  rap- 
idly destroyed  if  allowed  to  run  after  they  are  dull. 
Cutters  are  oftener  wasted  by  trying  to  cut  with  them 
when  they  are  dull  than  by  too  much  grinding.  Grind 
the  faces  radial  with  a  free  cutting  wheel.  Do  not  let 
the  wheel  become  glazed,  as  this  will  draw  the  temper 
of  the  cutter. 

In  Chapter  VI.  was  given  a  series  of  cutters  for  cut- 
ting gears  having  12  teeth  and  more.  Thus,  it  was 
there  implied  that  any  gear  of  same  pitch,  having  135 
teeth,  136  teeth,  and  so  on  up  to  the  largest  gears,  and, 
also,  a  rack,  could  be  cut  with  one  cutter.  If  this  cut- 
ter is  4  P,  we  would  cut  with  it  all  4  P  gears,  having 
135  teeth  or  more,  and  we  would  also  cut  with  it  a  4  P 
rack.  Now,  instead  of  always  referring  to  a  cutter  by 
the  number  of  teeth  in  gears  it  is  designed  to  cut,  it 
has  been  found  convenient  to  designate  it  by  a  letter 
or  by  a  number.  Thus,  we  call  a  cutter  of  4  P,  made 
to  cut  gears  135  teeth  to  a  rack,  inclusive,  No.  1,  4  P. 
We  have  adopted  numbers  for  designating  involute 
involute  Gear  gear-cutters  as  in  the  following  table : 

No.  1  will  cut  wheels  from  135  teeth  to  a  rack  inclusive. 
«     2         "  "  55       "      134  teeth     " 

«     3         «  "  35       "        54     "         " 

«     4         «  «  26       "        34     "         " 

u     5         tt  a  21       "        25     "         " 

«     e         «  «  17       «        20     lt         " 

»     7         «  «  14       «        1G     "         " 

«     g         «  «  12       "        13     "         " 


PROVIDENCE,    R.    I.  83 

By  this  plan  it  takes  eight  cutters  to  cut  all  gears 
having  twelve  teeth  and  over,  of  any  one  pitch. 

Thus  it  takes  eight  cutters  to  cut  all  involute  4  P 
gears  having  twelve  teeth  and  more.  It  takes  eight 
other  cutters  to  cut  all  involute  gears  of  5  P,  having 
12  teeth  and  more.  A  No.  8,  5  P  cutter  cuts  only  5  P 
gears  having  12  and  13  teeth.  A  No.  6,  10  P  cutter 
cuts  only  10  P  gears  having  17,  18,  19  and  20  teeth. 
On  each  cutter  is  stamped  the  number  of  teeth  at  the 
limits  of  its  range,  as  well  as  the  number  of  the  cutter. 
The  number  of  the  cutter  relates  only  to  the  number 
of  teeth  in  gears  that  the  cutter  is  made  for. 

In  ordering  cutters  for  involute  spur-gears  two  things 
must  be  given : 

1.  Either  the  number  of  teeth  to  be  cut  in  the  gear    HOW  to  order 
or  the  number  of  the  cutter,  as  given  in  the  foregoing ters- 

table. 

2.  Either  the  pitch  of  the  gear  or  the  diameter  and 
number  of  teeth  to  be  cut  in  the  gear. 

If  25  teeth  are  to  be  cut  in  a  6  P  involute  gear,  the 
cutter  will  be  No.  5,  6  P,  which  cuts  all  6  P  gears  from 
21  to  25  teeth  inclusive.  If  it  is  desired  to  cut  gears 
from  15  to  25  teeth,  three  cutters  will  be  needed,  No. 
5,  No.  6  and  No.  7  of  the  pitch  required.  If  the  pitch 
is  8  and  gears  15  to  25  teeth  are  to  be  cut,  the  cutters 
should  be  No.  5,  8  P,  No.  6,  8  P,  and  No.  7, 8  P. 

For  each  pitch  of  epicycloidal,  or  double-curve  gears,      Epicycioidai 
24  cutters  are  made.      In  coarse-pitch  gears,  the  varia-  curve  Cutters, 
tion  in  the  shape  of  spaces  between  gears  of  consecu- 
tive-numbered teeth  is  greater  than  in  fine-pitch  gears. 

A  set  of  cutters  for  each  pitch  to  consist  of  so  large 
a  number  as  24,  was  established  for  the  reason  that 
double  carve  teeth  were  formerly  preferred  in  coarse 
pitch  gears.  The  tendency  now,  however,  is  to  use  the 
involute  form. 

Our  double  curve  cutters  have  a  guide  shoulder  on  each 
side  for  the  depth  to  cut.  When  this  shoulder  just  reaches 
the  periphery  of  the  blank  the  depth  is  right.  The  marks 
which  these  shoulders  make  on  the  blank,  should  be  as  nar- 
row as  can  be  seen,  when  the  blanks  are  sized  right. 


84  BHOWN    &    SHAKPE    MFG.    CO. 

Double-curve  gear-cutters  are  designated  by  letters 
instead  of  by  numbers ;  this  is  to  avoid  confusion  in 
ordering. 

Following  is  the  list  of  epicycloidal  or  double-curve 
gear-cutters: — 
cySSKu!  iEpor  Cutter  A  cuts  12  teeth.     Cutter  M  cuts  27  to  29  teeth. 

Double -curve         lt       L>      «      -|  •>        tt  u        XT      it      on    "    ^       " 

Gear  Cutters.  »  rfu 

"  C  "  14  "  "  O  lt  34  "  37  " 

"  D  "  15  "  .       "  P  "  38  "  42  <• 

"  E  "  16  "  "  Q  "  43  "  49  " 

u  F  a  17  «  «  R  u  50  «  59  « 

«  G  "  18      "  "  S  "    60  "  74     " 

"  H  «  19      "  "  T  "     75  "  99     " 

«  i  a  20      "  "  II  "  100  ':  149  " 

«  J  "  21  to  22  «  V  "  150  «  249  « 

"  K  "  23  to  24  "  W  "  250  "  Back. 

"  L  "  24  to  26  u  X  "  Back. 

A  cutter  that  cuts  more  than  one  gear  is  made  of 
proper  form  for  the  smallest  gear  in  its  range.  Thus, 
cutter  J  for  21  to  22  teeth  is  correct  for  21  teeth; 
cutter  S  for  60  to  74  teeth  is  correct  for  60  teeth, 
and  so  on. 

Ep?cySoidai      *n  or(lermg  epicycloidal    gear-cutters  designate  the 

Cutters.  letter   of  the    cutter  as   in  the  foregoing  table,  also 

either  give  the  pitch  or  give  data  that  will  enable  us 
to  determine  the  pitch,  the  same  as  directed  for  invo- 
lute cutters. 

More  care  is  required  in  making  and  adjusting  epi- 
cycloidal gears  than  in  making  involute  gears. 

B^vV?  °Gear     *n  or(^ermg  bevel- gear  cutters  three  things  must  be 

Cutters.  given  : 

1.  The  number  of  teeth  in  each  gear. 

2.  Either   the  pitch   of  gears  or  the  largest  pitch 

diameter  of  each  gear;  see  Fig.  17. 

3.  The  length  of  tooth  face. 

If  the  shafts  are  not  to  run  at  right  angles,  it 
should  be  so  stated,  and  the  angle  given.  Involute 
cutters  only  are  used  for  cutting  bevel  gears.  No  at- 
tempt should  be  made  to  cut  epicyclodial  tooth  bevel  gears 
with  rotary  disk  cutters. 


PROVIDENCE,    K.    I.  85 


In  ordering  worm-wheel  cutters,  three  things  must    How  to  order 


be  given  :  Cutters. 

1.  Number  of  teeth  in  the  wheel. 

2.  Pitch  of  the  worm;    see   Chapter  XI. 

3.  Whole  diameter  of  worm. 

In  any  order  connected  with  gears  or  gear-cutters, 
when  the  word  "  Diameter  "  occurs,  we  usually  under- 
stand that  the  pitch  diameter  is  meant.  When  the 
whole  diameter  of  a  gear  is  meant  it  should  be  plainly 
written.  Care  in  giving  an  order  often  saves  the  delay 
of  asking  further  instructions.  An  order  for  one  gear- 
cutter  to  cut  from  25  to  30  teeth  cannot  be  filled,  be- 
cause it  takes  two  cutters  of  involute  form  to  cut  from 
25  to  30  teeth,  and  three  cutters  of  epicycloidal  form 
to  cut  from  25  to  30  teeth. 

Sheet  zinc  is  convenient  to  sketch  gears  upon,  and 
also  for  making  templets.  Before  making  sketch,  it  is 
well  to  give  the  zinc  a  dark  coating  with  the  following 
mixture  :  Dissolve  1  ounce  of  sulphate  of  copper  (blue 
vitriol)  in  about  4  ounces  of  water,  and  add  about  one- 
half  teaspoonful  of  nitric  acid.  Apply  a  thin  coating 
with  a  piece  of  waste. 

This  mixture  will  give  a  thin  coating  of  copper  to 
iron  or  steel,  but  the  work  should  then  be  rubbed  dry. 
Care  should  be  taken  not  to  leave  the  mixture  where  it 
is  not  wanted,  as  it  rusts  iron  and  steel. 

We  have  sometimes  been  asked  why  gears  are  noisy. 
Not  many  questions  can  be  asked  us  to  which  we  can 
give  a  less  definite  answer  than  to  the  question  why 
gears  are  noisy. 

We  can  indicate  only  some  of  the  causes  that  may 
make   gears  noisy,   such  as:  —  depth   of    cutting   not 
right  —  in  this  particular  gears  are  oftenercut  too  deep 
than  not  deep  enough  ;  (more   noise  may  be  caused 
by  cutting  the  driver  too  deep  than  by  cutting  the 
driven   too   deep;)     cutting    not    central  —  this  may 
make  gears  noisy  in  one  direction  when  they  are  quiet 
while  running  in  the  other  direction  ;  centre  distance 
not   right  —  if    loo    deep    the    outer    corners   of    the 
teeth  in  one  gear  may  strike  the  fillets  of  the  teeth 
in  the  other  gear  ;   shafts  not  parallel  ;    frame  of  the 


86 


BROWN    &    SHARPE    MFG.    CO. 


machine  of  such  a  form  as  to  give  off  sound  vibrations. 
Even  when  we  examine  a  pair  of  gears  we  cannot 
always  tell  what  is  the  matter. 


IMPROVED  29°  SCREW  THREAD  TOOL  GAUGE. 


ACME  STANDARD. 


DEPTH  OF  GEAR  TOOTH  GAUGES. 


^~v 

BROWN&SHAHPEMFG.CO.     (       ) 
PROVIDENCE.R.I..  ^~^ 


Depth  of  Gear  Tooth  Gauges  for  all  regular  pitches,  from  3  to 
48  pitch  inclusive,  are  carried  in  stock. 

One  Gauge  answers  for  each  pitch,  and  indicates  the  extreme 
depth  to  be  cut. 


PART    II. 


CHAPTER  I. 
TANGENT  OF  ARC  AND  ANGLE. 


In  PART  II.  we  shall  show  how  to  calculate 
of  the  functions  of  a  right-angle  triangle  from  a  table 
of  circular  functions,  the  application  of  these  calcula- 
tions in  some  chapters  of  PART  I.  and  in  sizing  blanks 
and  cutting  teeth  of  spiral  gears,  the  selection  of 
cutters  for  spiral  gears,  the  application  of  continued 
fractions  to  some  problems  in  gear  wheels  and  cutting 
odd  screw-threads,  etc.,  etc. 

A  Function  is  a  quantity  that  depends  upon  another 
quantity  for  its  value.      Thus  the  amount  a  workman 
earns  is  a  function  of  the  time  he  has  worked  and  of  fi]^ncti011  de" 
his  wages  per  hour. 


In  any  right- angle    triangle,  O  A  B,  we  shall,  for    Right  -  angle 
convenience,  call  the  two  lines   that  form   the  right 
angle  O  A  B  the  sides,  instead  of  base  and  perpen- 
dicular.     Thus  O  A  B,  being  the  right  angle  we  call 
the  line  O  A  a  side,  and  the  line  A  B  a  side  also. 

When  we  speak  of  the  angle  A  O  B,  we  call  the  line 
O  A  the  side  adjacent.     When  we  are  speaking  of  theside  ad3acent- 
angle  ABO  we  call  the  line   A  B  the  side  adjacent. 
The  line  opposite  the  right  angle  is  the  hypothenuse.  Hypothenuse. 

In%  the  following  pages  the  definitions  of  circular 
functions  are  for  angles  smaller  than  90°,  and  not 
strictly  applicable  to  the  reasoning  employed  in  ana- 
lytical trigonometry,  where  we  find  expressions  for 
angles  of  270°,  760°,  etc. 


88 

Tangent. 


BROWN   &    SHARPE    MFG.    CO. 

The  Tangent  of  an  arc  is  the  line  that  touches  it  at 
one  extremity  and  is  terminated  by  a  line  drawn  from 
the  center  through  the  other  extremity.  The  tangent 
is  always  outside  the  arc  and  is  also  perpendicular  to 
the  radius  which  meets  it  at  the  point  of  tangency. 


Fig.  47. 

Thus,  in  Fig.  47,  the  line  A  B  is  the  tangent  of  the  arc 
A  C.  The  point  of  tangency  is  at  A. 

An  angle  at  the  center  of  a  circle  is  measured  by  the 
arc  intercepted  by  the  sides  of  the  angle.  Hence  the 
tangent  A  B  of  the  arc  A  C  is  also  the  tangent  of  the 
angle  A  O  B. 

In  the  tables  of  circular  functions  the  radius  of  the 
arc  is  unity,  or,  in  common  practice,  we  take  it  as  one 
inch.  The  radius  O  A  being  1",  if  we  know  the  length 
of  the  line  or  tangent  A  B  we  can,  by  looking  in  a 
table  of  tangents,  find  the  number  of  degrees  in  the 
angle  A  O  B. 
TO  find  the  Thus,  if  A  B  is  2.25"  long,  we  find  the  angle  A  O  B 

Degrees  in  an 

Angle.  is  66°  very  nearly.     That  is,  having  found  that  2.2460 

is  the  nearest  number  to  2.25  in  the  table  of  tangents 
at  the  end  of  this  volume,  we  find  the  corresponding 
degrees  of  the  angle  in  the  column  at  the  left  hand  of 
the  table  and  the  minutes  to  be  added  at  the  top  of 
the  column  containing  the  2. 2460. 

The  table  gives  angles  for  every  10',  which  is  suf- 
ficient for  most  purposes. 


PBOVIDENCE,    B.    I.  89 

Now,  if  we  have  a  right-angle  triangle  with  an  angle 
the  same  as  O  A  B,  but  with  O  A  two  inches  long,  the 
line  A  B  will  also  be  twice  as  long  as  the  tangent  of 
angle  A  O  B,  as  found  in  a  table  of  tangents. 

Let  us  take  a  triangle  with  the  side  O  A  =  5"  long,  flnf  xat^epleD  £ 
and  the  side  A  B  =  8"  long ;  what  is  the  number  of  «£ees  in  an 
degrees  in  the  angle  A  O  B  ?  • 

Dividing  8"  by  5  we  find  what  would  be  the  length 
of  A  B  if  O  A  was  only  1"  long.  The  quotient  then 
would  be  the  length  of  tangent  when  the  radius  is  V 
long,  as  in  the  table  of  tangents.  8  divided  by  5  is 
1.6.  The  nearest  tangent  in  the  table  is  1.6003  and 
the  corresponding  angle  is  58°,  which  would  be  the 
angle  A  O  B  when  A  B  is  8"  and  the  radius  O  A  is  5" 
very  nearly.  The  difference  in  the  angles  for  tangents 
1.6003  and  1.6  could  hardly  be  seen  in  practice.  The 
side  opposite  the  required  acute  angle  corresponds  to 
the  tangent  and  the  side  adjacent  corresponds  to  the 
radius.  Hence  the  rule : 

To  find  the  tangent  of  either  acute  angle  in  a  right-  T^ge^d  tne 
angle  triangle  :  Divide  the  side  opposite  the  angle  by 
the  side  adjacent  the  angle  and  the  quotient  will  be 
the  tangent  of  the  angle.  This  rule  should  be  com- 
mitted to  memory.  Having  found  the  tangent  of  the 
angle,  the  angle  can  be  taken  from  the  table  of  tan- 
gents. 

The  complement  of  an  angle  is  the  remainder  after     Complement 
subtracting  the  angle  from  90°.     Thus  40°  is  the  com- 
plement of  50°. 

The  Cotangent  of  an  angle  is  the  tangent  of  the  cotangent, 
complement  of  the  angle.  Thus,  in  Fig.  47,  the  line 
A  B  is  the  cotangent  of  A  O  E.  In  right-angle  tri- 
angles either  acute  angle  is  the  complement  of  the 
other  acute  angle.  Hence,  if  we  know  one  acute  angle, 
by  subtracting  this  angle  from  90°  we  get  the  other 
acute  angle.  As  the  arc  approaches  90°  the  tangent 
becomes  longer,  and  at  90°  it  is  infinitely  long. 

The  sign  of  infinity  is  oo.     Tangent  90°  —  oo. 


90  BROWN    &    8HARPE    MFG.    CO. 

Angieayt>yUttne  By  a  table  of  tangents,  angles  can  be  laid  out  upon 
Tangent }  E^X-  sheet  zinc,  etc.  This  is  often  an  advantage,  as  it  is  not 
convenient  to  lay  protractor  flat  down  so  as  to  mark 
angles  up  to  a  sharp  point.  If  we  could  lay  off  the 
length  of  a  line  exactly  we  could  take  tangents  direct 
from  table  and  obtain  angle  at  once.  It,  however,  is 
generally  better  to  multiply  the  tangent  by  5  or  10 
and  make  an  enlarged  triangle.  If,  then,  there  is  a 
slight  error  in  laying  off  length  of  lines  it  will  not 
make  so  much  difference  with  the  angle. 

Let  it  be  required  to  lay  off  an  angle  of  14°  30'.  By 
the  table  we  find  the  tangent  to  be  .25861.  Multiply- 
ing .25861  by  5  we  obtain,  in  the  enlarged  triangle, 
1. 29305"  as  the  length  of  side  opposite  the  angle  14° 
30'.  As  we  have  made  the  side  opposite  five  times  as 
large,  we  must  make  the  side  adjacent  five  times  as 
large,  in  order  to  keep  angle  the  same.  Hence,  Fig. 
48,  draw  the  line  A  B  5"  long  ;  perpendicular  to  this 
line  at  A  draw  the  line  A  O  1.293"  long ;  now  draw  the 
line  O  B,  and  the  angle  ABO  will  be  14°  30'. 

If  special  accuracy  is  required,  the  tangent  can  be 
multiplied  by  10;  the  line  A  O  will  then  be  2.586"  long 
and  the  line  A  B  10"  long.  Remembering  that  the 
acute  angles  of  a  right-angle  triangle  are  the  comple- 
ments of  each  other,  we  subtract  14°  30'  from  90'  and 
obtain  75°  30'  as  the  angle  of  A  O  B. 

The  reader  will  remember  these  angles  as  occurring 
in  PART  I.,  Chapter  IV.,  and  obtained  in  a  different 
way.  A  semicircle  upon  the  line  O  B  touching  the 
extremities  O  and  B  will  just  touch  the  right  angle  at 
A,  and  the  line  O  B  is  four  times  as  long  as  O  A. 

Let  it  be  required  to  turn  a  piece  4"  long,  1"  diam- 
eter at  small  end,  with  a  taper  of  10°  one  side  with  the 
other ;  what  will  be  the  diameter  of  the  piece  at  the 
large  end  ? 

A  section,  Fig.  49,  through  the  axis  of  this  piece  is 
Diameter  Uof  *a  ^e  same  as  ^  we  a(lded  two  right-angle  triangles,  O 

i1  ^  ^  ^   and    °    ^'  ^  '  t0    a   straigkt   PieCG   A    A  B  B'»    *" 

'  wide  and  4"  long,  the  acute  angles  B  and  B'  being  5°, 
thus  making  the  sides  O  B  and  O'  B'  10°  with  each 
other. 


PROVIDENCE,    E.    I. 


48. 


Fig.  49. 


BEOWN    &    SHAEPE    MFG.    CO. 

The  tangent  of  5°  is  .08748,  which,  multiplied  by 
4  ,  gives  .34992"  as  the  length  of  each  line,  A  O  and 
A'  O',  to  be  added  to  1"  at  the  large  end.  Taking 
twice  .34992"  and  adding  to  V  we  obtain  1.69984"  as 
the  diameter  of  large  end. 

This  chapter  must  be  thoroughly  studied  before 
taking  up  the  next  chapters.  If  once  the  memory 
becomes  confused  as  to  the  tangent  and  sine  of  an 
angle,  it  will  take  much  longer  to  get  righted  than  it 
will  to  first  carefully  learn  to  recognize  the  tangent 
of  an  angle  at  once. 

If  one  knows  what  the  tangent  is,  one  can  tell  better 
the  functions  that  are  not  tangents. 


93 


CHAPTER  II. 

SINE— COSINE  AND  SECANT :     SOME  OF  THEIR  APPLICATIONS  IB 
MACHINE  CONSTRUCTION. 


The  Sine  of  an  arc  is  the  line  drawn  from  one 
extremity  of  the  arc  to  the  diameter  passing  through 
the  other  extremity,  the  line  being  perpendicular  to 
the  diameter. 

Another  definition  is  :  The  sine  of  an  arc  is  the  dis- 
tance of  one  extremity  of  the  arc  from  the  diameter, 
through  the  other  extremity. 

The  sine  of  an  angle  is  the  sine  of  the  arc  that 
measures  the  angle. 

In  Fig.  50  ,  A  C  is  the  sine  of  the  arc  B  C,  and  of 
the  angle  B  O  C.  It  will  be  seen  that  the  sine  is 
always  inside  of  the  arc,  and  can  never  be  longer  than 
the  radius.  As  the  arc  ap- 
proaches 90°,  the  sine  comes 
nearer  to  the  radius,  and  at  90° 
the  sine  is  equal  to  1,  or  is  the 
radius  itself.  From  the  defini- 
tion of  a  sine,  the  side  A  C, 
opposite  the  angle  A  O  C,  in 
any  right-angle  triangle,  is  the 
sine  of  the  angle  A  O  C,  when 
O  C  is  the  radius  of  the  arc. 


Fig.  50. 


Hence  the  rule :     In  any  right-angle  triangle,  the 
opposite  either  acute   angle,   divided  by  the  hypothe- 
nuse,  is  equal  to  the  sine  of  the  angle. 

The  quotient  thus  obtained  is  the  length  of  side 
opposite  the  angle  when  the  hypothenuse  or  radius  is 
unity.  The  rule  should  be  carefully  committed  to 
memory. 


Arc 


To    find 
Sine. 


the 


94 


BSOWN    &    GHARPE    MFG.    CO. 


Chord  of  an 
Arc. 


A  Chord  is  a  straight  line  joining  the  extremities  of 
an  arc,  and  is  twice  as  long  as  the  sine  of  half  the 
angle  measured  by  the  arc.  Thus,  in  Fig.  50,  the 
chord  F  C  is  twice  as  long  as  the  sine  A  C. 


findxth?chord?  chor(1 


Polygon. 


Fig.  51. 

Let  there  be  four  holes  equidistant  about  a  circle 
3"  in  diameter  —  Fig.  51  ;  what  is  the  shortest  distance 
between  two  holes?  This  shortest  distance  is  the 
,  which  is  twice  the  sine  of  the  angle  COB. 
The  angle  A  O  B  is  one-quarter  of  the  circle,  and 
C  O  B  is  one-eighth  of  the  circle.  360°,  divided  by 
8=45°,  the  angle  COB.  The  sine  of  45°  is  .70710, 
which  multiplied  by  the  radius  1.5",  gives  length  C  B  in  the 
circle,  3"  in  diameter,  as  1.0GOG5".  Twice  this  length  is 
the  required  distance  A  B=2.1213". 

When  a  cylindrical  piece  is  to  be  cut  into  any  num- 
ber of  sides,  the  foregoing  operation  can  be  applied  to 
obtain  the  width  of  one  side.  A  plane  figure  bounded 
by  straight  lines  is  called  a  polygon. 


PROVIDENCE,    E.    I. 


95 


When  the  outside  diameter  and  the  number  of  sides  of 
a  regular  polygon  are  given,  to  find  the  length  of 
one  of  the  sides :  Divide  360°  by  twice  the  number  of 
sides  ;  multiply  the  sine  of  the  quotient  by  the  outer 
diameter,  and  the  product  will  be  the  length  of  one  of 
the  sides. 

Multiplying  by  the  diameter  is  the  same  as  multi- 
plying by  the  radius,  and  that  product  again  by  2. 

The  Cosine  of  an  angle  is  the  sine  of  the  comple-  cosine, 
ment  of  the  angle. 

In  Fig.  50,  C  O  D  is  the  complement  of  the  angle 
A  O  C ;  the  line  C  E  is  the  sine  of  COD,  and  hence 
is  the  cosine  of  B  O  C.  The  line  O  A  is  equal  to  C  E. 
It  is  quite  as  well  to  remember  the  cosine  as  the  part 
of  the  radius,  from  the  center  that  is  cut  off  by  the 
sine.  Thus  the  sine  A  C  of  the  angle  A  O  C  cuts  off 
the  cosine  O  A.  The  line  O  A  may  be  called  the 
cosine  because  it  is  equal  to  the  cosine  C  E. 

In  any  right-angle  triangle,  the  side  adjacent  either 
acute  angle  corresponds  to  the  cosine  when  the 
hypothenuse  is  the  radius  of  the  arc  that  measures 
the  angle :  hence:  Divide  the  side  adjacent  the  acute  TO  nnd  the 

.  .,7  Cosine. 

angle  by  the  hypothenuse,  and  the  quotient  will  be  the 
cosine  of  the  angle. 

When  a  cylindrical  piece  is  cut  into  a  polygon  of 
any  number  of  sides,  a  table  of  cosines  can  be  used 
obtain  the  diameter  across  the  sides. 


90 


BROWN   &    SHARPS    MFG.    CO. 


Let  a  cylinder,  2"  diameter,  Fig.  53,  be  cut  six-sided : 
what  is  the  diameter  across  the  sides  ? 

The  angle  A  O  B,  at  the  center,  occupied  by  one  of 
these  sides,  is  one-sixth  of  the  circle,  =60°.  The 
cosine  of  one-half  this  angle,  30°,  is  the  line  C  O; 
twice  this  line  is  the  diameter  across  the  sides.  The 
cosine  of  30°  is  .86602,  which,  multiplied  by  2,  gives 
1.73204"  as  the  diameter  across  the  sides. 

Of  course,  if  the  radius  is  other  than  unity,  the  cosine 

should  be  multiplied  by  the  radius,  and  the  product 

again  by  2,  in  order  to  get  diameter  across  the  sides ; 

or  what  is  the  same  thing,  multiply  the  cosine  by  the 

whole  diameter  or  the  diameter  across  the  corners. 

zmeter f  across     Tlle  rule  f or  obtaining  the  diameter  across  sides  of 

skies  of  a  Poly-  regular  polygon,  when  the  diameter  across  corners  is 

given,    will   then  be:     Multiply   the  cosine   of  360° 

divided  by  twice  the  number  of  sides,  by  the  diameter 

across  corners,  and  the  product  will  be  the  diameter 

.  across  sides. 

Look  at  the  right-hand  column  for  degrees  of  the 
cosine,  and  at  bottom  of  page  for  minutes  to  add  to 
the  degrees. 

The  Secant  of  an  arc  is  a  straight  line  drawn  from 
the  center  through  one  end  of  an  arc,  and  terminated 
by  a  tangent  drawn  from  the  other  end  of  the  arc. 

Thus,  in  Fig.  53,  the  line  O  B  is  the  secant  of  the 
angle  COB. 

A  C  B 


Secant. 


Secant. 


Fly.  53. 

find  the  In  any  right-angle  triangle,  divide  the  hypothenuse 
by  the  side  adjacent  either  acute  angle,  and  the  quo- 
tient will  be  the  secant  of  that  angle. 


PROVIDENCE,    R.    I. 

That  is,  if  we  divide  the  distance  O  B  by  O  C,  in 
the  right-angle  triangle  COB,  the  quotient  will  be 
the  secant  of  the  angle  COB. 

The  secant  cannot  be  less  than  the  radius ;  it  in- 
creases as  the  angle  increases,  and  at  90°  the  secant  is 
infinity =00  . 

A  six-sided  piece  is  to  be  1£"  across  the  sides  ;  how 
large  must  a  blank  be  turned  before  cutting  the  sides  ?  JJj? r°8^ 
Dividing  360°  by  twice  the  number  of  sides,  we  have 
30°,  which  is  the  angle  COB.      The  secant  of  30°  is 
1.1547. 

The  radius  of  the  six-sided  piece  is  .75". 

Multiplying  the  secant  1.1547  by  .75",  we  obtain  the 
length  of  radius  of  the  blank  O  B ;  multiplying  again 
by  2,  we  obtain  the  diameter  1.732"  +  . 

Hence,  in  a  regular  polygon,  when  the  diameter 
across  sides  and  the  number  of  sides  are  given,  to  find 
diameter  across  corners :  Multiply  the  secant  of  360° 
divided  by  twice  the  number  of  sides,  by  the  diameter 
across  sides,  and  the  product  will  be  the  diameter 
across  corners. 

It  will  be  seen  that  the  side  taken  as  a  divisor  has 
been  in  each  case  the  side  corresponding  to  the  radius 
of  the  arc  that  subtends  the  angle. 

The  versed  sine  of  an  acute  angle  is  the  part  of 
radius  outside  the  sine,  or  it  is  the  radius  minus  the 
cosine.  Thus,  in  Fig.  50,  the  versed  sine  of  the  arc 
BC  is  AB.  The  versed  sine  is  not  given  in  the  tables 
of  circular  functions  :  when  it  is  wanted  for  any  angle 
less  than  90°  we  subtract  the  cosine  of  that  angle  from 
the  radius  1.  Having  it  for  the  radius  1,  we  can 
multiply  by  the  radius  of  any  other  arc  of  which  we 
may  wish  to  know  the  versed  sine. 

Fig.  54  is  a  sketch  of  a  gear  tooth  of  IP.  In 
measuring  gear  teeth  of  coarse  pitch  it  is  sometimes  a 
convenience  to  know  the  chordal  thickness  of  the 
tooth,  as  at  ATB,  because  it  may  be  enough  shorter 
than  the  regular  tooth-thickness  AHB,  or  t,  to  require 
attention.  It  may  be  also  well  to  know  the  versed 
sine  of  the  angle  B,  or  the  distance  H,  in  order  to  tell 
where  to  measure  the  chordal  thickness. 


BROWN   &    SHARPE    MFG.    CO. 


NO.    13.     AUTOMATIC   GEAR  CUTTING   MACHINE. 
FOR  SPUR  AND  BEVEL  GEARS. 


PROVIDENCE,    R.    I. 


99 


FRONT  VIEW. 


REAR  VIEW. 


GEAR   MODEL. 
Shows  combination  of  six  different  kinds  of  gears. 


100 


CHAPTER   III. 

APPLICATION  OF  CIRCULAR  FUNCTIONS— WHOLE  DIAMETER  OF 
BEVEL  GEAR  BLANKS— ANGLES  OF  BEVEL  GEAR  BLANKS. 


The  rules  given  in  this  chapter  apply  only  to  bevel 
gears  having  the  center  angle  c  O  i  not  greater  than  90°. 
To  avoid  confusion  we  will  illustrate  one  gear  only. 
The  same  rules  apply  to  all  sizes  of  bevel  gears.  Fig. 
55  is  the  outline  of  a  pinion  4  P,  20  teeth,  to  mesh  with 
a  gear  28  teeth,  shafts  at  right  angles.  For  making 
sketch  of  bevel  gears  see  Chapter  IX.,  PART!. 

In  Fig.  55,  the  line  O  m'  m  is  continued  to  the  line 

a  b.      The  angle  c'  O  i  that  the  cone  pitch-line  makes 

with  the  center  line  may  be  called  the  center  angle. 

Angle    of  The  center  angle  c  O  i  is  equal  to  the  angle  of  edge 

Edge.    Fig.  55.     ,.  ,    .  .      fi          .  -,  •.       ,,  ,  >, 

c  i  c.  c  ^  is  the  side  opposite  the  center  angle  c  O 
i,  and  c'  O  is  the  side  adjacent  the  center  angle,  c' 
i  —  2.5";  c'  O  =  3.5".  Dividing  2.5"  by  3.5"  we 
obtain  .71428"  +  as  the  tangent  of  c'  O  i.  In  the  table 
we  find  .71329  to  be  the  nearest  tangent,  the  corre- 
sponding angle  being  35°  30'.  35  J°,  then,  is  the  center 
angle  c'  O  *  and  the  angle  of  edge  c  i  n.  very  nearly. 

When  the  axes  of  bevel  gears  are  at  right  angles  the 
angle  of  edge  of  one  gear  is  the  complement  of  angle 
of  edge  of  the  other  gear.  Subtracting,  then,  35  J° 
from  90°  we  obtain  54J°  as  the  angle  of  edge  of  gear 
28  teeth,  to  mesh  with  gear  20  teeth,  Fig.  55,  from  which  we 
have  the  rule  for  obtaining  centre  angles  when  the  axes  of 
gears  are  at  right  angles. 

Divide  the  radius  of  the  pinion  by  the  radius  of  the  gear 
and  the  quotient  will  be  the  tangent  of  centre  angle  of  the 
pinion. 

Now  subtract  this  centre  angle  from  90  deg.  and  we  have 
the  centre  angle  of  the  gear. 

The  same  result  is  obtained  by  dividing  the  number  of 
teeth  in  the  pinion  by  the  number  of  teeth  in  the  gear ;  the 
quotient  is  the  tangent  of  the  centre  angle. 


PROVIDENCE,    K.    I. 


101 


Fig.  55. 

BEVEL  GEAR  DIAGRAM. 


102  BROWN    &    SHARPE    MFG.    CO. 

Angle  of  Face.  TO  obtain  angle  of  face  O  m"  c',  the  distance  c  O 
becomes  the  side  opposite  and  the  distance  m"  c  is 
the  side  adjacent. 

Tbe  distance  c  O  is  3L5",  the  radius  of  the  28  tooth 
bovol   ee&r.      The  distance  c  m"  is  by  measurement 


Dividing  3.5  by  2.82  we  obtain  1.2411  for  tangent 
of  angle  of  face  O  m"  c'.  The  nearest  tangent  in  the 
table  is  1.2422  and  the  corresponding  angle  is  51°  10'. 
To  obtain  cutting  angle  c  O  n"  we  divide  the  distance 
c'  n"  by  c  O.  By  measurement  c'  n'  is  2.2".  Divid- 
ing 2.2  by  3.5  we  obtain  .62857  for  tangent  of  cutting 
angle.  The  nearest  corresponding  angle  in  the  table 
is  32°10'. 

The  largest  pitch  diameter,  kj,  of  a  bevel  gear,  as  in 
Fig.  56,  is  known  the  same  as  the  pitch  diameter  of 
any  spur  gear.  Now,  if  we  know  the  distance  b  o  or 
its  equal  a  q,  we  can  obtain  the  whole  diameter  of 
bevel  gear  blank  by  adding  twice  the  distance  b  o  to 
the  largest  pitch  diameter. 

Twice  the  distance  b  o,  or  what  is  the  same  thing, 
the  sum  of  a  q  and  b  o  is  called  the  diameter  incre- 
ment, because  it  is  the  amount  by  which  we  increase 
the  largest  pitch  diameter  to  obtain  the  whole  or  out- 
side diameter  of  bevel  gear  blanks.  The  distance  b  o 
can  be  calculated  without  measuring  the  diagram. 

The  angle  b  o  j  is  equal  to  the  angle  of  edge. 

The  angle  of  edge,  it  will  be  remembered,  is  the 
angle  formed  by  outer  edge  of  blank  or  ends  of  teeth 
with  the  end  of  hub  or  a  plane  perpendicular  to  the 
axis  of  gear. 

The  distance  b  o  is  equal  to  the  cosine  of  angle  of 
edge,  multiplied  by  the  distance  j  o.  The  distance  j  o 
is  the  addendum,  as  in  previous  chapters  (=s). 

Hence  the  rule  for  obtaining  the  diameter  increment 
of  any  bevel  gear:  Multiply  the  cosine  of  angle  of 
edge  by  the  working  depth  of  teeth  (D"),  and  the 
product  will  be  the  diameter  increment. 

By  the  method  given  on  page  102  we  find  the  angle 
of  edge  of  gear  (Fig.   56)  is  56°  20'.     The  cosine 
of  56°  20°  is  .55436,  which,  multiplied  by  |",  or  the 
depth  of  the  3  P  gear,  gives  the  diameter  increment  of 
the  bevel  gear  18  teeth,  3  P  meshing  with  pinion  of  12 


PROVIDENCE,    E.    I. 


103 


104  BROWN    &    SHARPE    MFG.    CO. 

teeth.  I  of  .55436=.369"  +  (or  .37",  nearly).  Adding 
the  diameter  increment,  .37",  to  the  largest  pitch 
diameter  of  gear,  6",  we  have  6.37"  as  the  outside 
diameter. 

In  the  same  manner,  the  distance  c  d  is  half  the 
diameter  increment  of  the  pinion.  The  angle  c  d  k  is 
equal  to  the  center  angle  of  pinion,  and  when  axes  are 
at  right  angles  is  the  complement  of  center  angle  of 
gear.  The  center  angle  of  pinion  is  33°  40'.  The 
cosine,  multiplied  by  the  working  depth,  gives  .555" 
for,  diameter  increment  of  pinion,  and  we  have  4.555" 
for  outside  diameter  of  pinion. 

In  turning  bevel  gear  blanks,  it  is  sufficiently  accu- 
rate to  make  the  diameter  to  the  nearest  hundredth  of 
an  inch. 

Incre  ^e  sma^  angle  o  Oj  is  called  the  angle  increment. 
When  shafts  are  at  right  angles  the  face  angle  of  one 
gear  is  equal  to  the  center  angle  of  the  other  gear, 
minus  the  angle  increment. 

Thus  the  angle  of  face  of  gear  (Fig.  56)  is  less  than 
the  center  angle  D  O  &,  or  its  equal  Oj  k  by  the  angle 
o  Oj.  That  is,  subtracting  o  Oj  from  Oj  k,  the  re- 
mainder will  be  the  angle  of  face  of  gear. 

Subtracting  the  angle  increment  from  the  center 
angle  of  gear,  the  remainder  will  be  the  cutting 
angle. 

The  angle  increment  can  be  obtained  by  dividing 
oj,  the  side  opposite,  by  Oj,  the  side  adjacent,  thus 
finding  the  tangent  as  usual. 

The  length  of  cone-pitch  line  from  the  common 
center,  O  to  j,  can  be  found,  without  measuring  dia- 
gram, by  multiplying  the  secant  of  angle  Oj  k,  or  the 
center  angle  of  pinion,  by  the  radius  of  largest  pitch 
diameter  of  gear. 

The  secant  of  angle  O,/  k,  33°  40',  is  1.2015,  which, 
multiplied  by  3",  the  radius  of  gear,  gives  3.6045"  as 
the  length  of  line  O  j. 

Dividing  oj  by  OJ,  we  have  for  tangent  .0924,  and 
for  angle  increment  5°  20'. 

The  angle  increment  can  also  be  obtained  by  the 
following  rule : 


PROVIDENCE,    K.    I.  105 

Divide  the  sine  of  center  angle  by  half  the  num- 
ber of  teeth,  and  the  quotient  will  be  the  tangent  of 
increment  angle. 

Subtracting  the  angle  increment  from  center  angles 
of  gear  and  pinion,  we  have  respectively : 
Cutting  angle  of  gear,  51°. 
Cutting  angle  of  pinion,  28°  20'. 

.Remembering  that  when   the  shafts    are   at   right 
angles,  the  face  angle  of  a  gear  is  equal  to  the  cutting 
angle  of  its  mate  (Chapter  X.  part  1),  we  have: 
Face  angle  of  gear,  28°  20'. 
Face  angle  of  pinion,  51°. 

It  will  be  seen  that  both  the  whole  diameter  and  the 
angles  of  bevel  gears  can  be  obtained  without  making 
a  diagram.  Mr.  George  B.  Grant  has  made  a  table  of 
different  pairs  of  gears  from  1  to  1  up  to  10  to  1,  con- 
taining diameter  increments,  angle  increments  and 
centre  angles,  which  is  published  in  his  "Treatise  on 
Gears."  "Formulas  in  Gearing,"  published  by  us,  also 
contains  extensive  tables  for  bevel  gearing.  We  have 
adopted  the  terms  "diameter  increment,"  "angle  incre- 
ment," and  "centre  angle"  from  him.  He  uses  the 
term  "back  angle"  for  what  we  have  called  angle  of 
edge,  only  he  measures  the  angle  from  the  axis  of  the 
gear,  instead  of  from  the  side  of  the  gear,  or  from  theA£oJay outran 
end  of  hub,  as  we  have  done ;  that  is,  his  "back  angle  "sine. 
is  the  complement  of  our  angle  of  edge. 

In  laying  out  angles,  the  following  method  may  be 


Jf'iy.  57. 


1C6 


BROWN    &    SHARPE    MFG.    CO. 


Back 

Cone  Radius. 


preferred,  as  it  does  away  with  the  necessity  of  making 
aright  angle:  Draw  a  circle,  ABO  (Fig.  57),  ten 
inches  in  diameter.  Set  the  dividers  to  ten  times  the 
sine  of  the  required  angle,  and  point  off  this  distance 
in  the  circumference  as  at  A  B.  From  any  point  O  in 
the  circumference,  draw  the  lines  O  A  and  O  B.  The 
angle  A  O  B  is  the  angle  required.  Thus,  let  the  re- 
quired angle  be  12°.  The  sine  of  12°  is  .20791,  which, 
multiplied  by  10,  gives  2.0791",  or  2Tfg-"  nearly,  for 
the  distance  A  B. 

Any  diameter  of  circle  can  be  taken  if  we  multiply 
the  sine  by  the  diameter,  but  10"  is  very  convenient, 
as  all  we  have  to  do  with  the  sine  is  to  move  the 
decimal  point  one  place  to  the  right. 

If  either  of  the  lines  pass  through  the  centre,  then  the 
two  lines  which  do  not  pass  through  the  centre  will  form  a 
right  angle.  Thus,  if  O  B  passes  through  the  centre  then 
the  two  lines  A  B  and  A  0  will  form  a  ri^ht  ano-le  at  A. 


Na  =  No.  of  Teeth  in  Gear. 

•   Nb  =  No.  of  Teeth  in  Pinion. 

OC  =  Centre  Angle  of  Gear. 


Measure  the  back  cone  radius  a  b  for  the  gear,  or  b  c  for  the  pinion. 
This  is  equal  to  the  radius  of  a  spur  gear,  the  number  of  teeth  in  which 
would  determine  the  cutter  to  use.  Hence  twice  a  b  times  the  diametral 
pitch  equals  the  number  of  teeth  for  which  the  cutter  should  be  selected 
for  the  gear.  Looking  in  the  list  on  page  240  the  proper  number  for  the 
cutter  can  be  found. 

Thus,  let  the  back  cone  radius  a  b  be  4"  and  the  diameter  pitch  be  8. 
Twice  four  is  8  and  8  x  8  is  64,  from  which  it  can  be  seen  that  the  cutter 
must  be  of  shape  No.  2,  as  64  is  between  55  and  134,  the  range  covered  by 
a  No.  2  cutter. 

The  number  of  teeth  for  which  the  cutter  should  be  selected  can  also 
be  found  by  the  following  formula: 


Tan.  OC 


Na 


Nb 


No.  of  teeth  to  select  cutter  for  gear=        A          for  pinion  =  sin 

If  the  gears  are  mitres  or  are  alike,  only  one  cutter  is  needed;   if  one 
gear  is  larger  than  the  other,  two  may  be  needed. 


107 


CHAPTER  IV. 
SPIRAL  GEARS— CALCULATIONS  FOR  LEAD  OF  SPIRALS, 


When  the  teeth  of  a  gear  are  cut,  not  in  a  straight  8Piral  Gear< 
path,  like  a  spur  gear,  but  in  a  helical  or  screw- like 
path,  the  gear  is  called,  technically,  a  twisted  or  screw 
gear,  but  more  generally  among  mechanics,  a  spiral 
gear.  A  distinction  is  sometimes  made  between  a 
screw  gear  and  a  twisted  gear.  In  twisted  gears  the 
pitch  surfaces  roll  upon  each  other,  exactly  like  spur 
gears,  the  axes  being  parallel,  the  same  as  in  Fig.  1, 
Part  I.  In  screw  gears  there  is  an  end  movement, 
or  slipping  of  the  pitch  surfaces  upon  each  other,  the 
axes  not  being  parallel.  In  screw  gearing  the  action 
is  analogous  to  a  screw  and  nut,  one  gear  driving 
another  by  the  end  movement  of  its  tooth  path.  This 
is  readily  seen  in  the  case  of  a  worm  and  worm-wheel, 
when  the  axes  are  at  right  angles,  as  the  movement  of 
wheel  is  then  wholly  due  to  the  end  movement  of 
worm  thread.  But,  as  we  make  the  axes  of  gears  more 
nearly  parallel,  they  may  still  be  screw  gears,  but  the 
distinction  is  not  so  readily  seen. 

We  can  have  two  gears  that  are  alike  run  together, 
with  their  axes  at  right  angles,  as  at  A  B,  Fig.  59. 

The  same  gear  may  be  used  in  a  train  of  screw  gears 
or  in  a  train  of  twisted  gears.  Thus,  B,  as  it  relates  to 
A,  may  be  called  a  screw  gear ;  but  in  connection  with 
C,  the  same  gear,  B,  may  be  called  a  twisted  gear. 
These  distinctions  are  not  usually  made,  and  we  call 
all  helical  or  screw-like  gears  made  on  the  Universal 
Milling  Machine  spiral  gears. 

When  two  external  spiral  gears  run  together,  with     Direction  of 
their  axes  parallel,  the  teeth  of  the  erears  must  have  erence  to  Axes. 

.,      ,         j         .      ,  Fig.59. 

opposite  hand  spirals. 


108  BKOWN    &    SHARPE    MFG.    CO. 

Thus,  in  Fig.  59  the  gear  B  has  right  hand  spiral 
teeth,  and  the  gear  C  has  left  hand  spiral  teeth.  When 
the  axes  of  two  spiral  gears  are  at  right  angles,  both 
gears  must  have  the  same  hand  spiral  teeth.  A  and 
B,  Fig.  59,  have  right  hand  spiral  teeth.  If  both  gears 
A  and  B  had  left  hand  spiral  teeth,  the  relative  direc- 
tion in  which  they  turn  would  be  reversed. 

Spiral  Lead.  rpne  spiral  lead  or  lead  of  spiral  is  the  distance  the 
spiral  advances  in  one  turn.  A  cylinder  or  gear  cut 
with  spiral  grooves  is  merely  a  screw  of  coarse  pitch  or 
long  lend;  that  is,  a  spiral  is  a  coarse  lead  screw,  and 
a  screw  is  a  fine  lead  spiral. 

Since  the  introduction  and  extensive  use  of  the 
Universal  Milling  Machine,  it  has  become  customary 
to  call  any  screw  cut  in  the  milling  machine  a  spiral. 
The  spiral  lead  is  given  as  so  many  inches  to  one 
turn.  Thus,  a  cylinder  having  a  spiral  groove  that 
advances  six  inches  to  one  turn,  is  said  to  have  a  six 
inch  spiral. 

In  screws  the  pitch  is  often  given  as  so  many  turns 
to  one  inch.  Thus,  a  screw  of  -J"  lead  is  said  to  be  2 
turns  to  the  inch.  The  reciprocal  expression  is  not 
much  used  with  spirals.  For  example,  it  would  not 
be  convenient  to  speak  of  a  spiral  of  6"  lead,  as  -J-  turns 
to  one  inch. 

The  calculations  for  spirals  are  made  from  the  func- 
tions of  a  right  angle  triangle. 

the  ^ufc  ^rom  PaPer  a  light  angle  triangle,  one  side  of 
ri&nt  {ingle  6"  l°ng>  and  the  other  side  of  the 
right  angle  2".  Make  a  cylinder  6"  in  circumference. 
It  will  be  remembered  (Part  I.,  Chapter  II.)  that  the 
circumference  of  a  cylinder,  multiplied  by  .3183,  equals 
the  diameter— 6" x. 3183  =  1.9098".  Wrap  the  paper 
triangle  around  the  cylinder,  letting  the  2"  side  be 
parallel  to  the  axis,  the  6"  side  perpendicular  to  the 
axis  and  reaching  around  the  cylinder.  The  hypoth- 
eneuse  now  forms  a  helix  or  screw-like  line,  called 
a  spiral.  Fasten  the  paper  triangle  thus  wrapped 
around.  See  Fig.  60. 


PROVIDENCE,    n.    I. 


109 


B 


E 


FIG.  58 -RACKS  AND  GEARS, 


IROWN    &    SHARPE     MF'G.    CO. 


PR O V I DE  N CE ,    R.  I . 


FIG.  59.-SPIRAL  GEARING, 


110 


BROWN    &    SHARPE    MFG.    CO. 


parts  of 


LEAD  OF  SPIRAL 


/          TV  V- 

/    i        \ 


Fig.  60. 


If  we  now  turn  this  cylinder  A  B  C  D  one  turn  in 
the  direction  of  the  arrow,  the  spiral  will  advance  from 
0  to  E.  This  advance  is  the  lead  of  the  spiral. 

The  angle  EOF,  which  the  spiral  makes  with  the 
axis  E  0,  is  the  angle  of  the  spiral.  This  angle  is  found 
as  in  Chapter  I.  The  circumference  of  the  cylinder 
corresponds  to  the  side  opposite  the  angle.  The  pitch 
of  the  spiral  corresponds  to  the  side  adjacent  the  angle. 
Hence  the  rule  for  angle  of  spiral: 

Divide  the  circumference  of  the  cylinder  or  spiral 
^  fjie  number  of  inches  of  spiral  to  one  turn,  and  the 
quotient  will  be  the  tangent  of  angle  of  spiral. 

When  the  angle  of  spiral  and  circumference  are  given, 
to  find  the  lead  : 

Divide  the  circumference  by  the  tangent  of  angle,  and 
the  quotient  will  be  the  lead  of  the  spiral. 

When  the  angle  of  spiral  and  the  lead  or  pitch  of  spiral 
are  given,  to  find  the  circumference  : 

Multiply  the  tangent  of  angle  by  the  lead,  and  the 
product  ivill  be  the  circumference. 

When  applying  calculations  to  spiral  gears  the  angle 
is  reckoned  at  the  pitch  circumference  and  not  at  the 
outer  or  addendum  circle. 

It  wiH  be  seen  that  when  two  spirals  of  different 
diameters  have  the  same  lead  the  spiral  of  less  diame- 
ter will  have  the  smaller  angle.  Thus  in  Fig.  60  if  the 
paper  triangle  had  been  4"  long  instead  of  6"  the  diam- 
eter of  the  cylinder  would  have  been  1.27"  and  the 
angle  of  the  spiral  would  have  been  only  63J  degrees. 


Ill 


CHAPTER  V. 

EXAMPLES  IN  CALCULATION  OF  THE  LEAD  OF  SPIRAL— ANGLE  OF 

SPIRAL— CIRCUMFERENCE  OF  SPIRAL  GEARS— 

A  FEW  HINTS  ON  CUTTING, 


It  will  be  seen  that  the  rules  for  calculating  the  cir- 
cumference of  spiral  gears,  angle  and  the  lead  of  spiral 
are  the  same  as  in  Chapter  I.,  for  the  tangent  and  angle 
of  a  right  angle  triangle.  In  Chapter  IV.,  the  word 
"circumference"  is  substituted  for  "side  opposite," 
and  the  words  "lead  of  spiral"  are  substituted  for 
"side  adjacent." 

When  two  spiral  gears  are  in  mesh  the  angle  of 
spiral  should  be  the  same  in  one  gear  as  in  the  other, 
in  order  to  have  the  shafts  parallel  and  the  teeth  work 
properly  together.  When  two  gears  both  have  right 
hand  spiral  teeth,  or  both  have  left  hand  spiral  teeth, 
the  angle  of  their  shafts  will  be  equal  to  the  sum  of 
the  angles  of  their  spirals.  But  when  two  gears  have 
different  hand  spirals  the  angle  of  their  shafts  will  be 
equal  to  the  difference  of  their  angles  of  spirals. 
Thus,  in  Fig.  59  the  gears  A  and  B  both  have  right 
hand  spirals.  The  angle  of  both  spirals  is  45°,  their 
sum  is  90°,  or  their  axes  are  at  right  angles.  But  C 
has  a  left  hand  spiral  of  45°.  Hence,  as  the  difference 
between  angles  of  spirals  of  B  and  C  is  0,  their  axes 
are  parallel. 

If  two  45°  gears  of  the  same  diameter  have  the  same 
number  of  teeth  the  lead  of  the  spiral  will  be  alike  in 
both  gears:  if  one  gear  has  more  teeth  than  the  other 
the  lead  of  spiral  in  the  larger  gear  should  be  longer 
in  the  same  ratio.  Thus,  if  one  of  these  gears  has  50 
teeth,  and  the  other  has  25  teeth,  the  lead  of  spiral 
in  the  50  tooth  gear  should  be  twice  as  long  as  that  of  ent  diameters 
the  25  tooth  gear.  Of  course,  the  diameter  of  pitch 


112  BROWN    &    SHARPE    MFG.    CO. 

circle  should  be  twice  as  large  in  the  50  tooth  as  in  the 
25  tooth  gear. 

In  spirals  where  the  angle  is  45°  the  circumference 
is  the  same  as  the  spiral  lead,  because  the  tangent  of 
45°  is  1. 
variation  in     Sometimes  the  circumference  is  varied  to  suit  a  pitch 

Circumference 

to  suit  a  spiral,  that  can  be  cut  on  the  machine  and  retain  the  angle 
required.  This  would  apply  to  cutting  rolls  for  mak- 
ing diamond-shaped  impressions  where  the  diameter 
of  the  roll  is  not  a  matter  of  importance. 

When  two  gears  are  to  run  together  in  a  given 
velocity  ratio,  it  is  well  first  to  select  spirals  that  the 
machine  will  cut  of  the  same  ratio,  and  calculate  the 
numbers  of  teeth  and  angle  to  correspond.  This  will 
often  save  considerable  time  in  figuring. 

The  calculations  for  spiral  gears  present  no  special 
difficulties,  but  sometimes  a  little  ingenuity  is  required 
to  make  work  conform  to  the  machine  and  to  such 
cutters  as  we  may  have  in  stock. 

Let  it  be  required  to  make  two  spiral  gears  to  run 
with  a  ratio  of  4  to  1,  the  distance  between  centres  to 
be  3.125"  (3|"),  the  axes  to  be  parallel. 

By  rule  given  in  Chapter  XII.,  Part  I.,  we  find  the 
diameters  of  pitch  circles  will  be  5"  and  1£".  Let  us 
take  a  spiral  of  48"  lead  for  the  large  gear,  and  a 
spiral  of  12"  lead  for  the  small  gear.  The  circumfer- 
ence of  the  5"  pitch  circle  is  15.70796".  Dividing 
the  circumference  by  the  lead  of  the  spiral,  we  have 
is. 7^1.96  =t 32724"  for  tangent  of  angle  of  spiral.  In 
the  table  the  nearest  angle  to  tangent,  .32724",  is  18°  10'. 

As  before  stated,  the  angle  of  the  teeth  in  the  small 
gear  will  be  the  same  as  the  angle  of  teeth  or  spiral  in 
the  large  gear. 

this  rule  SiveS  the   anSle  at  the   Pitch   surface 

Upon  looking  at  a  small  screw  of  coarse  pitch, 
it  will  be  seen  that  the  angle  at  bottom  of  the  thread 
is  not  so  great  as  the  angle  at  top  of  thread;  that  is, 
the  thread  at  bottom  is  nearer  parallel  to  the  centre 
line  than  that  at  the  top. 

This  will  be  seen  in  Fig.  61,  where  A  0  is  the  centre 
line;  If  shows  direction  of  bottom  of  thread,  and  d  g 


PROVIDENCE,   E,   I. 


113 


shows  direction  of  top  of  thread.  The  angle  A/ b  is 
less  than  the  angle  A  g  d.  The  difference  of  angle 
being  due  to  the  warped  nature  of  a  screw  thread. 

A  cylinder  2"  diameter  is  to  have  spiral  grooves  20°  caf^fiS?on  of 
with  the  centre  line  of  cylinder;  what  will  be  the  leadLeadof  spiral. 
of  spiral?     The  circumference  is  6.2832".     The  tan- 
gent of  20°  is  .36397.     Dividing  the  circumference  by 
the  tangent  of  angle,  we  obtain  £;f Iffy  =  17. 2 6" -{-for 
lead  of  spiral. 


Fig.  61. 

In  Chapter  XI,  part  I,  it  is  stated  that,  when  gashing 
the  teeth  of  a  worm-wheel,  the  angle  of  the  teeth 
across  the  face  is  measured  from  the  line  parallel  to  the 
axis  of  the  wheel. 

To  obtain  this  angle  from  the  worm,  divide  the  lead 
by  the  pitch  circumference  of  the  worm,  and  the  quo- 
tient will  be  the  tangent  of  the  angle  of  the  thread 
with  a  perpendicular  to  the  axis. 


114 


CHAPTER  VI. 

NORMAL  PITCH  OF  SPIRAL  GEARS— CURVATURE  OF  PITCH 
SURFACE— FORM  OF  COTTERS. 


Curve?181  to  a     ^  Normal  to  a  curve  is  a  line  perpendicular  to  the 
tangent  at  the  point  of  tangency. 


In  Fig.  62,  the  line  B  C  is  tangent  to  the  arc  D  E  F, 
and  the  line  A  E  O,  being  perpendicular  to  the  tan- 
gent at  E  the  point  of  tangency,  is  a  normal  to  the 
arc. 

Fig.  63  is  a  representation  of  the  pitch  surface  of  a 
spiral  gear.  A'  D'  C'  is  the  circular  pitch,  as  in  Part 
I.  A  D  C  is  the  same  circular  pitch  seen  upon  the 
periphery  of  a  wheel.  Let  A  D  be  a  tooth  D  0  and  a 
space.  Now,  to  cut  this  space  D  C,  the  path  of  cut- 
ting is  along  the  dotted  line  a  b.  By  mere  inspection, 
we  can  see  that  the  shortest  distance  between  two 
teeth  along  the  pitch  surface  is  not  the  distance 
ADC. 

Let  the  line  A  E  B  be  perpendicular  to  the  sides  of 
teeth  upon  the  pitch  surface.  A  continuation  of  this 
line,  perpendicular  to  all  the  teeth,  is  called  the 
Normal  Helix.  The  line  A  E  B,  reaching  over  a 
tooth  and  a  space  along  the  normal  helix,  is  called  the 
Normal  Pitch,  or  the  normal  linear  pitch. 


PROVIDENCE,  R.  I. 


lid 


!A 


D' 


D    bC 


Fig.  63. 


116  BROWN    &    SHARPE    MFG.    CO. 

Normal  pitch.  The  Normal  Pitch  of  a  spiral  gear  is  then :  The 
shortest  distance  between  the  centers  of  two  consecutive 
teeth  measured  along  the  pitch  surface. 

In  spur  gears  the  normal  pitch  and  circular  pitch 
are  alike.  In  the  rack  D  D,  Fig.  58,  the  linear  pitch 
and  normal  pitch  are  alike. 

Cutter  for  From  the  foregoing  it  will  be  seen  that,  if  we  should 
cut  the  space  D  C  with  a  cutter,  the  thickness  of  which 
at  the  pitch  line  is  equal  to  one-half  the  circular  pitch, 
as  in  spur  wheels,  the  space  would  be  too  wide,  and 
the  teeth  would  be  too  thin.  Hence,  spiral  gears 
should  be  cut  with  thinner  cutters  than  spur  gears  of 
the  same  circular  pitch. 

The  angle  C  A  B  is  equal  to  the  angle  of  the  spiral. 
The  line  A  E  B  corresponds  to  the  cosine  of  the  angle 
CAB.  Hence  the  rule :  Multiply  the  cosine  of  angle 
TO  find  Nor-  o-f  spiral  by  the  circular  pitch,  and  the  product  will  be 
the  normal  pitch.  One-half  the  normal  pitch  is  the 
proper  thickness  of  cutter  at  the  pitch  line. 

If  the  normal  pitch  and  the  angle  are  known,  Divide 
the  normal  pitch  by  the  cosine  of  the  angle  and  the  quo- 
tient will  be  the  circular  pitch. 

This  may  be  required  in  a  case  of  a  spiral  pinion  run- 
ning in  a  rack.  The  perpendicular  to  the  side  of  the 
rack  is  taken  as  the  line  from  which  to  calculate  angle 
of  teeth.  That  is,  this  line  would  correspond  to  the 
axial  line  in  a  spiral  gear;  and,  when  the  axis  of  the 
gear  is  at  right  angles  to  the  rack,  the  angle  of  the 
teeth  with  the  side  of  the  rack  is  obtained  by  subtract- 
ing this  angle  from  90°. 

The  angle  of  the  rack  teeth  with  the  side  of  the 
rack  can  also  be  obtained  by  remembering  that  tho 
cosine  of  the  angle  of  spiral  is  the  sine  of  the  angle  of 
the  teeth  with  the  side  of  the  rack. 

The  addendum  and  working  depth  of  tooth  should 
correspond  to  the  normal  pitch,  and  not  to  the  circular 
pitch.  Thus,  if  the  normal  pitch  is  12  diametral,  the 
addendum  should  be  -fa",  the  thickness  .1309",  and  so 
on.  The  diameter  of  pitch  circle  of  a  spiral  gear  is 
calculated  from  the  diametral  pitch.  Thus  a  gear  of 
30  teeth  10  P  would  be  3"  pitch  diameter. 


PROVIDENCE,   B.   L  117 

But  if  the  normal  pitch  is  12  diametral  pitch,  the 
blank  will  be  3T2/  diameter  instead  of  3ry. 

It  is  evident  that  the  normal  pitch  varies  with  the  formal  Pitcn 
angle  of  spiral.  The  cutter  should  be  for  the  normal 
pitch.  In  designing  spiral  gears,  it  is  well  first  to  look 
over  list  of  cutters  on  hand,  and  see  whether  there  are 
cutters  to  which  the  gears  can  be  made  to  conform. 
This  may  avoid  the  necessity  of  getting  a  new  cutter, 
or  of  changing  both  drawing  and  gears  after  they  are 
under  way.  To  do  this,  the  problem  is  worked  the 
reverse  of  the  foregoing;  that  is: 

First  calculate  to  the  next  finer  pitch  cutter  than  ^make  An- 
would  be  required  for  the  diametral  pitch. 

Let  us  take,  for  example,  two  gears  10  pitch  and  30 
teeth,  spiral  and  axes  parallel.  Let  the  next  finer  cut- 
ter be  for  12  pitch  gears.  The  first  thing  is  to  find  the 
angle  that  will  make  the  normal  pitch.  .2618",  when  the 
circular  pitch  is  .3142".  See  table  of  tooth  parts. 
This  means  (Fig.  63)  that  the  line  A  D  0  will  be  .3142" 
when  A  E  B  is  .2618".  Dividing  .2618"  by  .3142"  (see 
Chap.  IV.)»  we  obtain  the  cosine  of  the  angle  CAB, 
which  is  also  the  angle  of  the  spiral,  ;ff  Jf  =.833. 

The  same  quotient  comes  by  dividing  10  by  12, 
•f|  ==.833  -f  ;  that  is,  divide  one  pitch  by  the  other,  the 
larger  number  being  the  divisor.  Looking  in  the  table, 
we  find  the  angle  corresponding  to  the  cosine  .833  is 
33°  30'.  We  now  want  to  find  the  pitch  of  spiral  that 
will  give  angle  of  33  J°  on  the  pitch  surface  of  the  wheel, 
3"  diameter.  Dividing  the  circumference  by  the  tan- 
gent of  angle,  we  obtain  the  pitch  of  spiral  (see  Chap. 
V.)  The  circumference  is  9.4248".  The  tangent  of 
33°  30'  is  .66188,  5-:|ff|^=14.23 ;  and  we  have  for 
our  spiral  14.23"  lead. 

When  the  machine  is  not  arranged  for  the  exact    when   exact 

Pitch  cannot  be 

pitch  of  spiral  wanted,  it  is  generally  well  enough  to  cut. 
take  the  next  nearest  spiral.  A  half  of  an  inch  more 
or  less  in  a  spiral  10"  pitch  or  more  would  hardly  be 
noticed  in  angle  of  teeth.  It  is  generally  better  to 
take  the  next  longer  spiral  and  cut  enough  deeper  to 
bring  center  distances  right.  When  two  gears  of  the 
same  size  are  in  mesh  with  their  axes  parallel,  a  change 


118 


BROWN    &    SHARPE    MFG.    CO. 


of  angle  of  teeth  or  spiral  makes  no  difference  in  the 
correct  meshing  of  the  teeth. 

-^l1^  wnen  gears  of  different  size  are  in  mesh,  due 
sizes  of  Mesh,  regard  must  be  had  to  the  spirals  being  in  pitch,  pro- 
portional to  their  angular  velocities  (see  Chapter  V.) 

We  come  now  to  the  curvature  of  cutters  for  spiral 
gears;  that  is,  their  shape  as  to  whether  a  cutter  is 
made  to  cut  12  teeth  or  100  teeth.  A  cutter  that  is  right, 
Shape  of  cut-  to  cut  a  spur  gear  3"  diameter,  may  not  be  right  for  a 
spiral  gear  3"  diameter.  To  find  the  curvature  of 
cutter,  fit  a  templet  to  the  blank  along  the  line  of  the 
normal  helix,  as  A  E  B,  letting  the  templet  reach  over 
about  one  normal  pitch.  The  curvature  of  this  templet 
will  be  nearer  a  straight  line  than  an  arc  of  the  adden- 
dum circle.  Now  find  the  diameter  of  a  circle  that  will 
approximately  fit  this  templet,  and  consider  this  circle 
as  the  addendum  circle  of  a  gear  for  which  we  are  to 
select  a  cutter,  reckoning  the  gear  as  of  a  pitch  the 
same  as  the  normal  pitch. 


Fig.  64. 

Thus,  in  Fig.  64,  suppose  the  templet  fits  a  circle 
3£"  diameter,  if  the  normal  pitch  is  12  to  inch,  dia- 
metral, the  cutter  required  is  for  12  P  and  40  teeth. 
The  curvature  of  the  templet  will  not  be  quite  circular, 
but  is  sufficiently  near  for  practical  purposes.  Strictly, 


PROVIDENCE,    E.   I.  119 

a  flat  templet  cannot  be  made  to  coincide  with  the 
normal  helix  for  any  distance  whatever,  but  any  greater 
refinement  than  we  have  suggested  can  hardly  be  car- 
ried out  in  a  workshop. 

This  applies  more  to  an  end  cutter,  for  a  disk  cutter 
may  have  the  right  shape  for  a  tooth  space  and  still 
round  off  the  teeth  too  much  on  account  of  the  warped 
nature  of  the  teeth. 

The  difference  between  normal  pitch  and  linear  or 
circular  pitch  is  plainly  seen  in  Figs  58  and  59. 

The  rack  D  D,  Fig.  58,  is  of  regular  form,  the  depth 
of  teeth  being  -J--J-  of  the  circular  pitch,  nearly  (.6866  of 
the  pitch,  accurately).  If  a  section  of  a  tooth  in  either 
of  the  gears  be  made  square  across  the  tooth,  that  is  a 
normal  section ,  the  depth  of  the  tooth  will  have  the 
same  relation  to  the  thickness  of  the  tooth  as  in  the 
rack  just  named. 

But  the  teeth  of  spiral  gears,  looking  at  them  upon 
the  side  of  the  gears,  are  thicker  in  proportion  to  their 
depth,  as  in  Fig.  59.  This  difference  is  seen  between 
the  teeth  of  the  two  racks  D  D  and  E  E,  Fig.  58.  In 
the  rack  D  D  we  have  20  teeth,  while  in  the  rack  E  E 
we  have  but  14  teeth ;  yet  each  rack  will  run  with  each 
of  the  spiral  gears  A,  B  or  C,  Fig.  59,  but  at  different 
angles. 

The  teeth  of  one  rack  will  accurately  fit  the  teeth  of 
the  other  rack  face  to  face,  but  the  sides  of  one  rack 
will  then  be  at  an  angle  of  45°  with  the  sides  of  the 
other  rack.  At  F  is  a  guide  for  holding  a  rack  in  mesh 
with  a  gear. 

The  reason  the  racks  will  each  run  with  either  of  the 
three  gears  is  because  all  the  gears  and  racks  have  the 
same  normal  pitch.  When  the  spiral  gears  are  to  run 
together  they  must  both  have  the  same  normal  pitch. 
Hence,  two  spiral  gears  may  run  correctly  together 
though  the  circular  pitch  of  one  gear  is  not  like  the 
circular  pitch  of  the  other  gear. 

If  a  rack  is  to  run  at  any  angle  other  than  90°  with 
the  axis  of  the  gear  it  is  well  to  determine  the  data 
from  a  diagram,  as  it  is  very  difficult  to  figure  the 
angles  and  sizes  of  the  teeth  without  a  sketch  or 
diagram. 


120 


CHAPTER  VII. 
CUTTING  SPIRAL  GEARS  IN  A  UNIVERSAL  MILLING  MACHINE. 


A  rotary  disk  cutter  is  generally  preferable  to  a  shank 
cutter  or  end  mill  on  account  of  cutting  faster  and  hold- 
ing its  shape  longer.  In  cutting  spiral  grooves,  it  is 
sometimes  necessary  to  use  an  end  mill  on  account  of 
the  warped  character  of  the  grooves,  but  it  is  very  sel- 
dom necessary  to  use  an  end  mill  in  cutting  spiral  gears. 
setSSSnof  the  Before  cutting  into  a  blank  it  is  well  to  make  a  slight 
Machine.  trace  of  the  spiral  with,  the  cutter,  after  the  change 
gears  are  in  place,  to  see  whether  the  gears  are  correct. 
If  the  material  of  the  gear  blanks  is  quite  expensive,  it 
is  a  safe  plan  to  make  trial  blanks  of  cast  iron  in  order 
to  prove  the  setting  of  the  machine,  before  cutting  into 
die  expensive  material. 

The  cutting  of  spiral  gears  may  develop  some  curi- 
ous facts  to  one  that  has  not  studied  warped  surfaces. 
The  gears,  Fig.  59,  were  cut  with  a  planing  tool  in  a 
shaper,  the  spiral  gear  mechanism  of  a  Universal  Mill- 
ing Machine  having  been  fastened  upon  the  shaper. 
The  tool  was  of  the  same  form  as  the  spaces  in  the  nick 
D  D,  Fig.  58.  All  spiral  gears  of  the  same  pitch  can  be 
cut  in  this  manner  with  one  tool.  The  nature  of  this 
cutting  operation  can  be  understood  from  a  considera- 
tion of  the  meshing  of  straight  side  rack  teeth  with  a 
spiral  gear,  as  in  Fig.  58.  Spiral  gears  that  run  cor- 
rectly with  a  rack,  as  in  Fig.  58,  will  run  correctly 
with  each  other  when  their  axes  are  parallel,  as  at  B  C, 
Fig.  59;  but  it  is  not  considered  that  they  are  quite 
correct,  theoretically,  to  run  together  when  the  gears 
have  the  same  hand  spiral,  and  their  axes  are  at  right 


PKOVIDENCE,   E.   I. 


121 


Fig.  65 


Fig.  66 


122 


BROWN    &    SHARPS    MFG.    CO. 


angles,  as  AB,  Fig.  59,  though  they  will  run  well  enough 
practically.  The  operation  of  cutting  spiral  teeth  with 
a  planer  tool  is  sometimes  c'd\\cd planing  the  teeth.  Plan- 
ing is  an  accurate  way  of  shaping  teeth  that  are  to  mesh 
with  rack  teeth  and  for  gears  on  parallel  shafts;  this 
method  has  been  employed  to  cut  spiral  pinions  that 
drive  planer  tables,  but  has  not  been  found  available 
for  general  use. 

It  is  convenient  to  have  the   data  of  spiral   gears 
arranged  as  in  the  following  table  : 


Gear. 

Pinion. 

No.  of  Teeth        .                         . 
Pitch  Diameter    .                         . 
Outside  Diameter 
Circular  Pitch      .                          . 
Angle  of  Teeth  with  Axis 
Normal  Circular  Pitch 
Pitch  of  Cutter  .                          . 
Addendum  s                                  . 
Thickness  of  Tooth   t 
Whole  Depth  D"+f  . 
No.  of  Cutter       .                         . 
Exact  Lead  of  Spiral 
Approximate  Lead  of  Spiral    . 

Gears  on  Milling  Machine  to  Cut  Spiral 
Gear  on  Worm     
1st  Gear  on  Stud          .... 
2nd  Gear  on  Stud        .... 
Gear  on  Screw     

Data. 


A  spiral  of  any  angle  to  45°  can  generally  be  cut  in 
a  Universal  Milling  Machine  without  special  attach- 
ments, the  cutter  being  at  the  top  of  the  work.  The 
cutter  is  placed  on  the  arbor  in  such  position  that  it 
can  reach  the  work  centrally  after  the  table  is  set  to 
the  angle  of  the  spiral.  In  order  to  cut  central,  it  is 
generally  well  enough  to  place  the  table,  before  setting 
it  to  the  angle,  so  that  the  work  centres  will  be  central 
with  the  cutter,  then  swing  the  table  and  set  it  to  the 
angle  of  the  spiral. 

For  very  accurate  work,  it  is  safer  to  test  the  posi-tl  Central  set- 
tion  of  the  centres  after  the  table  has  been  set  to  the 
angle. 


PROVIDENCE,    K.    I.,    U.    S.    A. 


123 


Tig.  67. 

USE  OF  VERTICAL    SPINDLE   MILLING   ATTACHMENT 
IN   CUTTING   SPIRAL   GEARS. 


1^1  BROWN    &    SHARPE    MFG.    CO. 

This  can  be  done  with  a  trial  piece,  Fig.  65,  which 
is  simply  a  round  arbor  with  centre  holes  in  the  ends. 
It  is  mounted  between  the  centres,  and  the  knee  is 
raised  until  the  cutter  sinks  a  small  gash,  as  at  A. 
This  gash  shows  the  position  of  the  cutter;  and  if  the 
gash  is  central  with  the  trial  piece,  the  cutter  will  be 
central  with  the  work.  If  preferred,  the  arbor  can  be 
dogged  to  the  work  spindle ;  and  the  line  B  0  drawn 
on  the  side  of  the  arbor  at  the  same  height  as  the  cen- 
tres; the  work  spindle  should  then  be  turned  quarter 
way  round  in  order  to  bring  the  line  at  the  top.  The 
gash  A  can  now  be  cut  and  its  position  determined  with 
the  line. 

In  cutting  small  gears  the  arbor  can  be  dogged  to  the 
work  spindle ;  the  distance  between  the  gear  blank  and 
the  dog  should  be  enough  to  let  the  dog  pass  the  cutter 
arbor  without  striking. 

A  spiral  gear  is  much  more  likely  to  slip  in  cutting 
than  a  spur  gear. 

For  gears  more  than  three  or  four  inches  in  diameter 
it  is  well  to  have  a  taper  shank  arbor  held  directly  in 
the  work  spindle,  as  shown  in  Figs.  67  and  68 ;  and  for 
the  heaviest  work,  the  arbor  can  be  drawn  into  the  spin- 
dle with  a  screw  in  a  threaded  hole  in  the  end  of  the 
shank. 

After  cutting  a  space  the  work  can  be  dropped  away 
from  the  cutter,  in  order  to  avoid  scratching  it  when 
coming  back  for  another  cut.  Some  workmen  prefer 
not  to  drop  the  work  away,  but  to  stop  the  cutter  and 
turn  it  to  a  position  in  which  its  teeth  will  not  touch 
the  work.  To  make  sure  of  finding  a  place  in  the  cut- 
ler that  will  not  scratch,  a  tooth  has  sometimes  been 
tiken  out  of  the  cutter,  but  this  is  not  recommended. 
The  safest  plan  is  to  drop  the  work  away. 

Angiegreater  In  cutting  spiral  gears  of  greater  angle  than  45°,  a 
vertical  spindle  milling  attachment  is  available,  as 
shown  in  Figs.  67  and  68. 

In  Fig.  67  the  cutter  is  at  90°  with  the  work  spindle 
when  the  table  is  set  to  0,  so  that  the  proper  angle  at 
which  the  table  should  be  set,  is  the  difference  between 
the  angle  of  the  spiral  and  90°.  Thus,  to  cut  a  70° 


PROVIDENCE;  R.  i.,  u.  s.  A. 


125 


Fig.  68. 

USE  OF  VERTICAL   SPINDLE   MILLING   ATTACHMENT 
IN   CUTTING   SPIRAL  GEARS 


126  BROWN    £    SHAKPE    MFG.    CO. 

spiral,  we  subtract  70°  from  90°,  and  the  remainder, 
20°,  is  the  angle  to  set  the  table.  In  cutting  on  the 
top,  Fig.  67,  the  attachment  is  set  to  0. 

In  Fig.  68  the  cutter  is  at  the  side  of  the  work ;  the 
table  is  set  to  0,  and  the  attachment  is  set  to  the  differ- 
ence between  90°  and  the  required  angle  of  spiral. 

In  setting  the  cutter  central  it  is  convenient  to  have  a 
small  knee  as  at  K,  Fig.  66.  A  line  is  drawn  upon  the 
knee  at  the  same  height  as  at  the  centres.  The  cutter 
arbor  is  brought  to  the  angle  as  just  shown,  and  a  gash 
is  cut  in  the  knee.  When  the  gash  is  central  with  the 
line,  the  cutter  will  be  central  with  the  work. 

The  cutter  can  be  set  to  act  upon  either  side  of  the 
gear  to  be  cut,  according  as  a  right  hand  or  a  left  hand 
spiral  is  wanted.  The  setting  in  Fig.  68  is  for  a  right 
hand  spiral. 

If  the  gear  blank  were  brought  in  front  of  the  cut- 
ter, and  the  reversing  gear  set  between  two  change 
gears,  the  machine  would  be  set  for  a  left  hand  spiral. 

For  coarser  pitches  than  about  12  P  diametral,  it  is 
well  to  cut  more  than  once  around,  the  finishing  cut 
being  quite  light  so  as  to  cut  smooth. 


127 


CHAPTER   VIII. 
SCREW  GEARS  AND  SPIRAL  GEARS— GENERAL  REMARKS. 


The  working  of  spiral  gears,  when  their  axes  are    working  of 

-,,.,,  .  Spiral  Gears. 

parallel,  is  generally  smoother  than  spur  gears.  A 
tooth  does  not  strike  along  its  whole  face  or  length  at 
once.  Tooth  contact  first  takes  place  at  one  side  of  the 
gear,  passes  across  the  face  and  ceases  at  the  other 
side  of  the  gear.  This  action  tends  to  cover  defects  in 
shape  of  teeth  and  the  adjustment  of  centres. 

Since  the  invention  of  machines  for  producing  accu- 
rate epicyloidal  and  involute  curves,  it  has  not  so  often 
been  found  necessary  to  resort  to  spiral  gears  for 
smoothness  of  action.  A  greater  range  can  be  had  in 
the  adjustment  of  centers  in  spiral  gears  than  in  spur 
gears.  The  angle  of  the  teeth  should  be  enough,  BO 
that  one  pair  of  teeth  will  not  part  contact  at  one  side 
of  the  gears  until  the  next  pair  of  teeth  have  met  on  the 
other  side  of  the  gears.  When  this  is  done  the  gears 
will  be  in  mesh  so  long  as  the  circumferences  of  their 
addendum  circles  intersect  each  other.  This  is  some- 
times necessary  in  gears  for  rolls. 

Relative  to  spur  and  bevel  gears  in  Part  I.,  Chapter 
XII.,  it  was  stated  that  all  gears  finally  wore  them- 
selves out  of  shape  and  might  become  noisy.  Spiral 
gears  may  be  worn  out  of  shape,  but  the  smoothness 
of  action  can  hardly  be  impaired  so  long  as  there  are 
any  teeth  left.  For  every  quantity  of  wear,  of  course, 
there  will  be  an  equal  quantity  of  backlash,  so  that  if 
gears  have  to  be  reversed  the  lost  motion  in  spiral 
gears  will  be  as  much  as  in  any  gears,  and  may  be 
more  if  there  is  end  plav  of  the  shafts.  In  spiral  gears  End  Pressure 

.,  .  ,  .,         ,      ..       ,  ,    .,      upon  Shafts  of 

there  is  end  pressure  upon  the  shafts,  because  01  tne  Spiral  Gears, 
screw-like  action  of  the  teeth.     This  end  pressure  is 
sometimes  balanced  by  putting  two  gears  upon  each 
shaft,  one  of  right  and  one  of  left  hand  spiral. 


128  BKOWN    &    SHARPE    MFG.    CO. 

The  same  result  is  obtained  in  solid  cast  gears  by 
making  the  pattern  in  two  parts — one  right  and  one 
left-hand  spiral.  Such  gears  are  colloquially  called 
"herring-bone  gears." 

In  an  internal  spiral  gear  and  its  pinion,  the  spirals 
of  both  wheels  are  either  right-handed  or  left-handed. 
Such  a  combination  would  hardly  be  a  mercantile 
product,  although  interesting  as  a  mechanical  feat. 

In  screw  or  worm-gears  the  axes  are  generally  at 
right  angles,  or  nearly  so.  The  distinctive  features  of 
screw  gearing  may  be  stated  as  follows : 

The  relative  angular  velocities  do  not  depend  upon 

the  diameters   of    pitch- cylinders,   as   in    Chapter  I., 

Distinctive  Part  I.     Thus  the  worm  in  Chapter  XL,  Fig.  35,  can 

features    of 

Screw  Gearing,  be  any  diameter — one  inch  or  ten  inches — without 
affecting  the  velocity  of  the  worm-wheel.  Conversely  if  the 
axes  are  not  parallel  we  can  have  a  pair  of  spiral  or  screw 
gears  of  the  same  diameter,  but  of  different  numbers  of 
teeth.  The  direction  in  which  a  worm-wheel  turns  depends 
upon  whether  the  worm  has  a  right-hand  or  left-hand  thread. 
When  angles  of  axes  of  worm  and  worm-wheel  are 
oblique,  there  is  a  practical  limit  to  the  directional 
relation  of  the  worm-wheel.  The  rotation  of  the 
worm-wheel  is  made  by  the  end  movement  of  the 
worm-thread. 

The  term  worm  and  worm-wheel,  or  worm-gearing, 
is  applied  to  cases  where  the  worms  are  cut  in  a  lathe, 
and  the  shapes  of  the  threads  or  teeth,  in  axial  section, 
are  like  a  rack.  The  shape  usually  selected  is  like  the 
rack  for  a  single  curve  or  involute  gear.  See  Chap. 
IV.,  Parti.  Worms  are  sometimes  cut  in  a  milling 
machine. 

If  the  form  of  the  teeth  in  a  pair  of  screw  gears  is 
determined  upon  the  normal  helix,  as  in  Chap.  VL, 
the  gears  are  usually  called  Spiral  Gears. 

If  we  let  two  cylinders  touch  each  other,  their  axes 
being  at  right  angles,  the  rotation  of  one  cylinder  will 
have  no  tendency  to  turn  the  other  cylinder,  as  in 
Chapter  L,  Part  I. 


PROVIDENCE,    R.    I.  129 


We  can  now  see  why  worms  and  worm-wheels 
out  faster  than  other  gearing.     The  length  of  worm-sofast- 
thread,  equal  to  more  than  the  entire  circumference  of 
worm,  comes  in  sliding  contact  with  each  tooth  of  the 
wheel  during  one  turn  of  the  wheel. 

The  angle  of  a  worm-thread  can  be  calculated  the 
same  as  the  angle  of  teeth  of  spiral  gear  ;  only,  the 
angle  of  a  worm  thread  is  measured  from  a  line  or 
plane  that  is  perpendicular  to  the  axis  of  the  worm. 


130 


CHAPTER    IX. 

CONTINUED  FRACTIONS— SOME  APPLICATIONS   IN    MACHINE 
CONSTRUCTION. 


a  Dcon\1nue0d     ^-  con^nue(i  fraction  is  one    that  has  unity  for  its 
Fraction.          numerator,  and  for  its  denominator  an  entire  number 
plus  a  fraction,  which  fraction  has  also  unity  for  its 
numerator,  and  for  its  denominator  an  entire  number 
plus  a  fraction,  and  thus  in  order. 
The  expression,  1  _  . 

9+1 

5   is  called  a  continued  frac- 

tion.    By  the  use  of  continued  fractions,  we  are  ena- 

ofPraconSnued  ^^e^  *°  ^n^  a  ^rac^on  expressed  in  smaller  numbers, 

Fractions.         that,  for  practical  purposes,  may  be  sufficiently  near  in 

value  to  another  fraction  expressed  in  large  numbers. 

If  we  were  required  to  cut  a  worm  that  would  mesh 

with  a  gear  4  diametral  pitch  (4  P.),  in  a  lathe  having 

3  to  1-inch  linear  leading  screw,  we  might,  without 
continued   fractions,  have   trouble  in  finding  change 
gears,  because    the    circular   pitch  corresponding  to 

4  diametral  pitch  is    expressed  in  large   numbers  : 


This  example  will  be  considered  farther  on.  For 
illustration,  we  will  take  a  simpler  example. 

"What  fraction  expressed  in  smaller  numbers  is  near- 

est in  value  to  y2^?      Dividing  the  numerator  and  the 

denominator  of  a  fraction  by  the  same  number  does 

not  change  the  value  of  the  fraction.     Dividing  both 

c?naS?ueiStems  of   T¥TT  by  29>  we  have   5^;  or>   wkat  is  the 


same  thing  expressed  as  a  continued  fraction,  5-i-^y  The 


continued  fraction  *+jfo  is  exactly  equal  to  -f/^.      If 
now,  we  reject  the  ^,  the  fraction  ^  will  be  larger 
than  6+  i  ,  because  the  denominator  has  been  dimin- 
ish* d,  5   being   less  than   5YV-      £  is  something  near 
expressed    in    smaller   numbers    than   29   for   a 


PROVIDENCE,    R.    I.  131 

numerator  and  146  for  a  denominator.  Keducing  \ 
and  T2j9^-  to  a  common  denominator,  we  have  ^=ij$ 
and  r24V=TM'  Subtracting  one  from  the  other,  we 
have  TJ¥,  which  is  the  difference  between  -J-  and  T2^-. 
Thus,  in  thinking  of  T%  as  -J-,  we  have  a  pretty  fair 
idea  of  its  value. 

There  are  fourteen  fractions  with  terms  smaller  than 
29  and  146,  which  are  nearer  -ffa  than  -J-  is,  such  as 
•J-jj-,  -J-J-  and  so  on  to  f}T.  In  this  case  by  continued  frac- 
tions we  obtain  only  one  approximation,  namely  -J,  and 
any  other  approximations,  as  -^f,  -J-f-,  &c.,  we  find  by 
trial.  It  will  be  noted  that  all  these  approximations 
are  smaller  in  value  than  •££$.  There  are  cases,  how- 
ever, in  which  we  can,  by  continued  fractions,  obtain 
approximations  both  greater  and  less  than  the  required 
fraction,  and  these  "will  be  the  nearest  possible  approxi- 
mations that  there  can  be  in  smaller  terms  than  the 
given  fraction. 

In  the  French  metric  system,  a  millimetre  is  equal 
to  .03937  inch;  what  fraction  in  smaller  terms  ex- 
presses .03937"  nearly?  .03937,  in  a  vulgar  fraction, 
is  yiHHHhj-.  Dividing  both  numerator  and  denominator 
by  3937,  we  have  ^sjjp..  Rejecting  from  the  de- 
nominator of  the  new  fraction,  •£-££?•»  the  fraction  -fa 
gives  us  a  pretty  good  idea  of  the  value  of  .03937". 
If  in  the  expression,  25+iiiA,  we  divide  both  terms  of 
the  fraction  -J-f -J4  ^y 1575,  the  value  will  not  be  changed. 
Performing  the  division,  we  have  * 

25  +  1 


2  +  787 
1575- 

We  can  now  divide  both  terms  of  fgfc  by  787, 
without  changing  its  value,  and  then  substitute  the 
new  fraction  for  -pffe  in  the  continued  fraction. 

Dividing  again,  and  substituting,  we  have : 
i 

25  +  1 


2  +  1 


2+  1 
787 

as   the   continued  fraction   that   is   exactly  equal   to 
.03937. 


132  BROWN    &    SHAKPE    MFG.    CO. 

In  performing  the  divisions,  the  work  stands  thus  : 


3937)  100000  (25 
7874 
21260 
19685 

1575)  3937  (2 
3150 

787)  1575  (2 
1574 

1)  787  (787 
787 

•o- 

That  is,  dividing  the  last  divisor  by  the  last  remain- 
der, as  in  finding  the  greatest  common  divisor.  The 
quotients  become  the  denominators  of  the  continued 
fraction,  with  unity  for  numerators.  The  denominators 
25,  2,  and  so  on,  are  called  incomplete  quotients,  since 
they  are  only  the  entire  parts  of  each  quotient.  The 
first  expression  in  the  continued  fraction  is  ^  or 
.04 — a  little  larger  than  .03937.  If,  now,  we  take 
i,  we  shall  come  still  nearer  .03937.  The  expres- 


sion  25^  is  merely  stating  that  1  is  to  be  divided  by 
25£.  To  divide,  we  first  reduce  25J  to  an  improper 
fraction,  ^-,  and  the  expression  becomes  ir,  or  one 
divided  by  ^.  To  divide  by  a  fraction,  "Invert  the 
divisor,  and  proceed  as  in  multiplication."  We 
then  have  -£T  as  the  next  nearest  fraction  to  .03937. 
-g2T— .0392  + ,  which  is  smaller  than  .03937.  To  get  still 
nearer,  we  take  in  the  next  part  of  the  continued  frac- 
tion, and  have 


4-1 


2  +  1 
2' 

We  can  bring  the  value  of  this  expression  into  a 
fraction,  with  only  one  number  for  its  numerator  and 
one  number  for  its  denominator,  by  performing  the 
operations  indicated,  step  by  step,  commencing  at  the 
last  part  of  the  continued  fraction.  Thus,  2  +  -J-,  or 
2£,  is  equal  to  f  ,  Stopping  here,  the  continued  frac- 
tion would  become  J  _ 


_ 
5 
T- 

1  1  _ 

Now,   j>     equals  \  ,  and  we  have  25  +  2  .      25f  equals 

2  5 

-1  ;  substituting  again,  we  have  jjh.      Dividing  1  by 
-1,   we   have  T|T.      T|T  is   the   nearest   fraction  to 


PROVIDENCE,    R.    I.  133 

.03937,  unless  we  reduce  the  whole  continued  fraction 
1 

25  +  1 

2  +  l_ 

2  +  *_,  which  would  give  us  back  the  .03937  itself. 

787 

TfT=.03937007,  which  is  only  ^^^^  larger 
.03937.  It  is  not  often  that  an  approximation  will 
come  so  near  as  this. 

This  ratio,  5  to  127,  is  used  in  cutting  millimeter     Practical  use 
thread  screws.     If  the  leading  screw  of  the  lathe  isExampie. 
1  to  one  inch,  the  change  gears  will  have  the  ratio  of 
5  to  127;  if  8  to  one  inch,  the  ratio  will  be  8  times 
as  large,  or  40  to  127;  so  that  with  leading  screw  8  to 
inch,  and  change  gears  40  and  127,  we  can  cut  milli- 
meter threads  near  enough  for  practical  purposes. 

The   foregoing   operations   are  more  tedious  in  de- 
scription than  in  use.     The  steps  have  been  carefully 
noted,   so   that  the  reason  for  each  step  can  be  seen 
from  rules  of  common  arithmetic,  the  operations  being 
merely  reducing  complex  fractions.     The  reductions, 
"sV>  ~?T>  T!T>  e^c-'  are  called,  conver gents,  because  they 
come  nearer  and  nearer  to  the  required  .03937.     The 
operations  can  be  shortened  as  follows: 

Let  us  find  the  fractions  converging  towards  .7854",  Example, 
the  circular  pitch  of  4  diametral  pitch,  .7854=T7Tr8Tr5/^; 
reducing  to  lowest  terms,  we  have  ftHHr      Applying 
the  operation  for  the  greatest  common  divisor: 

3927)  5000  (1 
3927 

1073)  3927  (3 
3219 

708)  1073  (1 
_708 

£165)  708  (1 
365 

843)  365  (1 
343 
22)  343  (15 

*L 

123 
110 
13)  22  (1 

I3 

9)  13  (1 
j) 

4)  9  (2 
8 

"1)  4  (4 
0 

Bringing  the  various  incomplete  quotients  as  de- 
nominators in  a  continued  fraction  as  before,  we  have: 


134  BROWN    &    SHARPE    MFG.    CO. 

1  _ 

1  +  1 

3  +  1 

1  +  1  _ 

1  +  1 


_ 

15  +  1 

1  +  1  _ 
1+1 


Now  arrange  each  partial  quotient  in  a  line,  thus  : 

13111       15        11         2  4 

1    I    i    1    tt    m    III    tt*    TST¥T 


Now  place  under  the  first  incomplete  quotient  the 
first  reduction  or  convergent  ^,  which,  of  course,  is  1  ; 
put  under  the  next  partial  quotient  the  next  reduction  or 
convergent  j-^-r  or  JT,  which  becomes  f  . 

1  is  larger  than  .7854,  and  f  is  less  than  .7854. 

Having  made  two  reductions,  as  previously  shown, 
we  can  shorten  the  operations  by  the  following  rule  for  next 
convergents:  Multiply  the  numerator  of  the  convergent 
just  found  "by  the  denominator  of  the  next  term  of  the  con- 
tinued fraction,  or  the  next  incomj)lete  quotient,  and  add 
to  the  product  the  numerator  of  the  preceding  convergent; 
the  sum  ivill  be  the  numerator  of  the  next  convergent. 

Proceed  in  the  same  way  for  the  denominator,  that 

is  multiply  the  denominator  of   the   convergent  just 

found  by  the  next  incomplete  quotient  and  add  to  the 

product  the  denominator  of  the  preceding  convergent  ; 

^the  sum  will  be  the  denominator  of  the  next  convergent. 

Continue  until  the  last  convergent  is  the  original  frac- 

tion.   Under  each  incomplete  quotient  or  denominator 

from  the  continued  fraction  arranged  in  line,  will  be 

seen  the  corresponding  convergent  or  reduction.     The 

convergent  -j-J-  is  the  one  commonly  used  in  cutting 

racks  4  P.     This  is  the  same  as  calling  the  circumference  of 

a  circle  22-7  when  the  diameter  is  one  (1)  ;  this  is  also  the 

common  ratio  for  cutting  any  rack.     The  equivalent  decimal 

to  1£  is  .7857  X,  being  about  y^f^  large.     In  three  set- 

tings for  rack  teeth,  this  error  would  amount  to  about  .001" 

For  a  worm,  this  corresponds  to  |f  threads  to  1"  ; 
now,  with  a  leading  screw  of  lathe  3  to  1",  we  would 
want  gears  on  the  spindle  and  screw  in  a  ratio  of  33 
to  14. 

Hence,  a  gear  on  the  spindle  with  66  teeth,  and  a 
gear  on  the  3  thread  screw  of  28  teeth,  would  enable 
us  to  cut  a  worm  to  fit  a  4  P  gear. 


135 


CHAPTER   X. 
ANGLE    OF    PRESSURE, 


In  Fig.  69,  let  A  be  any  flat  disk  lying  upon  a  hori- 
zontal plane.  Take  any  piece,  B,  with  a  square  end, 
a  b.  Press  against  A  with  the  piece  B  in  the  direction 
of  the  arrow. 


Fig.  69. 


Fig.  70. 


It  is  evident  A  will  tend  to  move  directly  ahead  of 
B  in  the  normal  line  c  d.  Now  (Fig.  70)  let  the  piece 
B,  at  one  corner/,  touch  the  piece  A.  Move  the  piece 
B  along  the  line  d  e,  in  the  direction  of  the  arrow. 

It  is  evident  that  A  will  not  now  tend  to  move  in 
the  line  d  6,  but  will  tend  to  move  in  the  direction  of 
the  normal  c  d.  When  one  piece,  not  attached,  presses 
against  another,  the  tendency  to  move  the  second 
piece  is  in  the  direction  of  the  normal,  at  the  point  of 
contact.  This  normal  is  called  the  line  of  pressure. 
The  angle  that  this  line  makes  with  the  path  of  the 
impelling  piece,  is  called  the  angle  of  pressure. 

In  Part  I.,  Chapter  IV.,  the  lines  B  A  and  B  A'  are 
called  lines  of  pressure.  This  means  that  if  the  gear 


Line  of  Press. 


136  BROWN    &    SHAEPE    MFG.    CO. 

drives  the  rack,  the  tendency  to  move  the  rack  is  not 
in  the  direction  of  pitch  line  of  rack,  but  either  in  the 
direction  B  A  or  B  A',  as  we  turn  the  wheel  to  the  left 
or  to  the  right. 

The  same  law  holds  if  the  rack  is  moved  in  the 
direction  of  the  pitch  line ;  the  tendency  to  move  the 
wheel  is  not  directly  tangent  to  the  pitch  circle,  as  if 
driven  by  a  belt,  but  in  the  direction  of  the  line  of 
pressure.  Of  course  the  rack  and  wheel  do  move  in 
the  paths  prescribed  by  their  connections  with  the 
framework,  the  wheel  turning  about  its  axis  and  the 
rack  moving  along  its  ways.  This  pressure,  not  in  a 
direct  path  of  the  moving  piece,  causes  extra  friction 
in  all  toothed  gearing  that  cannot  well  be  avoided. 

Although  this  pressure  works  out  by  the  diagram, 
as  we  have  shown,  yet,  in  the  actual  gears,  it  is  not  at 
all  certain  that  they  will  follow  the  law  as  stated, 
because  of  the  friction  of  teeth  among  themselves.  If 
the  driver  in  a  train  of  gears  has  no  bearing  upon  its 
tooth-flank,  we  apprehend  there  will  be  but  little 
tendency  to  press  the  shafts  apart. 
Arc  of  Action.  Tne  arc  through  which  a  wheel  passes  while  one  of 

its  teeth  is  in  contact  is  called  the  arc  of  action. 
temasof°imerl     Until  within  a  few  years,  the  base  of  a  system  of 
Geara g  e  a  b  L  e  double-curve  interchangeable  gears  was  12  teeth.     It 
is  now  15  teeth  in  the  best  practice  (see  Chapter  VII., 
Part  I.) 

The  reason  for  this  change  was  :  the  base,  15  teeth, 
gives  less  angle  of  pressure  and  longer  arc  of  contact, 
and  henco  longer  lifetime  to  gearis. 


137 


CHAPTER    XI. 
INTERNAL    GEARS. 


In  Part  L,  Chapter  VIII.,  it  is  stated  that  the  space 
of  an  internal  gear  is  the  same  as  the  tooth  of  a  spur 
gear.  This  applies  to  involute  or  single-curve  gears  as 
well  ns  to  double-curve  gears. 

The  sides  of  teeth  in  involute  internal  gears  are 
hollowing.  It,1  however,  has  been  customary  to  cut 
internal  gears  with  spur  gear-cutters,  a  No.  1  cutter 
generally  being  used.  This  makes  the  teeth  sides 
convex.  Special  cutters  should  be  made  for  coarse  Special  cut. 

..,..,,  T      ..      .       .  ters  for  coarse 

pitch  double-curve  gears.  In  designing  internal  gears,  Pitch, 
it  is  sometimes  necessary  to  depart  from  the  system 
with  15-tooth  base,  so  as  to  have  the  pinion  differ  from 
the  wheel  by  less  than  15  teeth.  The  rules  given  in 
Part  I.,  Chapters  VII.  and  VIII.,  will  apply  in  making 
gears  on  any  base  besides  15  teeth.  If  the  base  is 
low-numbered  and  the  pinion  is  small,  it  may  be  neces- 
sary to  resort  to  the  method  given  at  the  end  of  Chap- 
ter VII.,  because  the  teeth  may  be  too  much  rounded 
at  the  points  by  following  the  approximate  rules. 
The  base  must  be  as  small  as  the  difference  between  Base  f°r  in- 

.  .    .  *i  ternal      Gear 

the  internal  gear  and   its  pinion.     The  base  can  be  Teeth, 
smaller  if  desired. 

Let  it  be  required  to  make  an  internal  gear,  and 
pinion  24  and  18  teeth,  3  P.  Here  the  base  cannot 
be  more  than  6  teeth. 

In  Fig.  71  the  base  is  6  teeth.  The  arcs  A  K  and 
O  &,  drawn  about  T,  have  a  radius  equal  to  the  radius 
of  the  pitch  circle  of  a  6-tooth  gear,  3  P,  instead  of  a 
15-tooth  gear,  as  in  Chapter  VIII.,  Part  I. 

The  outline  of  teeth  of   both  gears  and  pinion  is    Description  of 
made  similar  to  the  gear  in  Chapter  VIII.      The  same 


138 


BROWN   &    SHARPS   MFG.    CO. 


GEAR,  24  TEETH. 
PINION,  18  TEETH,  3  P. 

P  =  3 

N=24  and  18 

P'=  1.0472* 

t=-    5236" 

8=     .3333" 

D";=     .6666* 

8+/=     .3857* 

P"+/=     7190" 


Fig.  71. 


INTERNAL   GEAR  AND    PINION    IN    MESH, 


PKOVIDENCE,    R.    I. 

letters  refer  to  similar  parts.  The  clearance  circle  is, 
however,  drawn  on  the  outside  for  the  internal  gear. 
As  before  stated,  the  spaces  of  a  spur  wheel  become 
the  teeth  of  an  internal  wheel.  The  teeth  of  internal 
gears  require  but  little  for  fillets  at  the  roots ;  they 
are  generally  strong  enough  without  fillets.  The 
teeth  of  the  pinion  are  also  similar  to  the  gear  in 
Chapter  VIII.,  substituting  6-tooth  for  15-tooth  base. 
To  avoid  confusion,  it  is  well  to  make  a  complete 
sketch  of  one  gear  before  making  the  other.  The  arc 
of  action  is  longer  in  internal  gears  than  in  external 
gears.  This  property  sometimes  makes  it  necessary 
to  give  less  fillets  than  in  external  gears. 

In  Fig.  71  the  angle  K  T  A  is  30°  instead  of  12°,  as 
in  Fig.  12.  This  brings  the  line  of  pressure  L  P  at 
an  angle  of  60°  with  the  radius  C  T,  instead  of  78°. 
A  system  of  spur  gears  could  be  made  upon  this 
6-tooth  base.  These  gears  would  interchange,  but  no 
gear  of  this  6-tooth  system  would  mesh  with  a  double- 
curve  gear  made  upon  the  15-tooth  system  in  Part  1. 


139 


140 


CHAPTER  XII. 


STRENGTH  OF  GEARING, 


We  have  been  unable  to  derive  from  onr  own  experi- 
ence, any  definite  rule  on  this  subject  but  would  refer 
those  interested  to  "Kent's  Mechanical  Engineers' 
Pocket  Book/'  where  a  good  treatment  of  the  subject 
can  be  found. 

We  give  a  few  examples  of  average  breaking  strain 
of  our  Combination  Gears,  as  determined  by  dyna- 
mometer, the  pressure  being  measured  at  the  pitch  line. 
These  gears  are  of  cast  iron,  with  cut  teeth. 


Diametral  Pitch. 

No.  Teeth. 

Revolutions 
per 
Minute. 

Pressure  at 
Pitch  Line. 

Face. 

10 

1  1-16 

110 

27 

1060 

8 

1  1-4 

72 

40 

1460 

6 

1  9-16 

72 

27 

2220 

5 

1  7-8 

90 

18 

2470 

These  are  the  actual  pressures  for  the  particular 
widths  given. 

If  we  take  a  safe  pressure  at  1-3  of  the  foregoing 
breaking  strain,  we  shall  have  for 

10  Pitch  353  1-3  Lbs.  at  the  Pitch  Line. 
8       "     486  2-3 

6,     "     740  "  " 

5       "     823  1-3  " 

The  width  of  the  face  of  a  gear  is  in  good  proportion 
when  it  is  2-j-  times  the  circular  pitch. 


PROVIDENCE,    II.    I. 


141 


TOOTH  PARTS. 


Fig.  73. 

GEAR  TOOTH  1    P 


142  BROWN    &   SHARPE    MFG.    CO. 

The  dimensions  of  tooth  parts  as  given  in  the  tables, 
pages  144  to  147,  are  correct  according  to  the  definition 
of  tooth  parts,  pages  4  and  16  ;  but,  as  the  pitch  line 
of  gears  is  curved,  the  thickness  of  a  tooth  will  not  be 
measured  on  the  pitch  line  if  the  caliper  is  set  to  the 
figures  given  in  the  tables  mentioned.  To  measure  the 
teeth  accurately  on  the  pitch  line,  the  caliper  must  be 
set  to  the  chordal  thickness  and  the  depth  setting  to  the 
pitch  line  must  be  to  the  corrected  «,  as  explained  and 
tabulated.  If  the  gear  blank  is  not  of  the  correct 
diameter,  the  proper  allowance  must  be  made  in  setting 
the  caliper  for  the  depth.  It  is  utterly  useless  to  be 
guided  by  the  outside  of  a  gear  blank  when  the  outside 
diameter  is  not  right.  The  measuring  of  the  tooth 
thickness  is  well  enough,  as  a  check,  but  it  is  oftentimes 
as  well  first  to  make  sure  that  the  spaces  are  cut  to  the 
right  depth. 

Fig.  73  is  a  sketch  of  a  gear  tooth  of  1  P.  In  meas- 
uring gear  teeth  of  coarse  pitch  accurately  the  chordal 
thickness  of  the  tooth,  ATB,  must  be  known,  because 
it  may  be  enough  shorter  than  the  regular  tooth-thick- 
ness AHB,  or  t,  to  require  attention.  It  may  be  also 
well  to  know  the  versed  sine  of  the  angle  /?',  or  the  dis- 
tance H,  in  order  to  tell  where  to  measure  the  chordal 
thickness. 

Chordal  Thicknesses  of  Teeth  of  Gears,  on  a 

Basis  of  1  Diametral  Pitch. 
N  —  Number  of  teeth  in  gears. 

T  =  Chordal  thickness  of  Tooth.         T  =  D'  sin.  ft' 
H  =  Height  of  Arc.  H  =  R  (1— cos.  ft') 

D'=  Pitch  Diameter. 
H  =  Pitch  Radius. 
ft'  =  90°  divided  by  the  number  of  teeth. 

NOTE. — For  any  pitch  not  in  the  following  tables  to  find 
corresponding  part : — Multiply  the  tabular  value  for  one  inch 
by  the  circular  pitch  required,  and  the  product  will  be  the 
value  for  the  pitch  given. 

Example  :  What  is  the  value  of  s  for  4  inch  circular  pitch  ? 
,3183  =  s  for  1"  P'  and  .3183  X  4  =  1.2732  =  s  for  4"  P'. 

The  expression  "Addendum  and  -^  "  (addendum  and  the 
module)  means  the  distance  of  a  tooth  outside  of  pitch  line 
and  also  the  distance  occupied  for  every  tooth  upon  the  diam- 
eter of  pitch  circle. 


PROVIDENCE,  R.  I. 


143 


CHORDAL  THICKNESSES  OF  TEETH  OF  GEARS. 

INVOLUTE. 


Cutter. 

T 

H 

Corrected 
S  for  Gear. 

No.  i  ^-135  T  —   P 

1.5707 

.0047 

1.0047 

No.  2  —  55  T  —  P 

1.5706 

.OI  I  2 

I.OII2 

No.  3  —  35  T  —  P 

1.5702 

.0176 

1.0176 

No.  4  —  26  T  —  P 

1.5698 

.0237 

1.0237 

No.  5  —  21  T—  P 

1.5694 

.0294 

1.0294 

No.  6—  17  T—  P 

1.5686 

.0362 

1.0362 

No.  7  —  14  T  —  P 

I-5675 

.0440 

1.0440 

No.  8  —  12  T  —  P 

1.5663 

.05H 

1.0514 

ii  T  —  P 

1.5654 

•0559 

1-0559 

10  T  —  P 

I-5643 

.O6l6 

1.  0616 

9T-  P 

1.5628 

.0684 

1  .0684 

8T—  P 

1.5607 

.0769 

1.0769 

EPICYCLOIDAL. 


Cutter. 

T 

H 

Corrected 
S  for  Gear. 

A—  12  T—  P 

1-5663 

.0514 

1.0514 

B  —  13  "—   P 

1.5670 

.0474 

1.0474 

C  —  14  "  —  P 

J-5675 

.0440 

1  .0440 

D—  15  "—  P 

1-5679 

.0411 

1.0411 

E_  16  T—  P 

1-5683   . 

.0385 

1-0385 

F  —  17  T  —  P 

1.5686 

.0362 

1.0362 

G_  18  "  —  P 

1.5688 

.0342 

1.0342 

H—  19"—  P 

1.5690 

.0324 

1.0324 

I  —  20  "  —   P 

1.5692 

.0308 

1.0308 

J  —  21"—   P 

1.5694 

.0294 

1.0294 

"K  —  23  T  —  P 

1.5696 

.0268 

1.0268 

L  _  25  T—  P 

1.5698 

.0247 

1.0247 

M—  27  T—  P 

1-5699 

.0228 

1.0228 

N  —  30  T  —  P 

1.5701 

.0208 

1  .0208 

0  -  34  T  —  P 

1-5703 

.Ol8l 

1.0181 

p  _  38  T  —  P 

1-5703 

.0162 

1.0162 

Q-43T-  P 

1-5705 

.0143 

1.0143 

R  —  50  T  —  P 

I-5705 

.0123 

1.0123 

S  _  60  T  —  P 

1.5706 

.OIO2 

I.OIO2 

T  _  75  T  -  P 

1-5707 

.0083 

1  .0083 

U  —loo  T  —  P 

1-5707 

.0060 

1  .0060 

V  —150  T  —  P 

1-5707 

.0045 

I.C045 

W—  250  T  —  P 

1.5708 

.0025 

I.OO25 

SPECIAL. 


No.  Teeth. 

T 

H 

Corrected 
S  for  Gear. 

9T—   i  P 
10  T  —  i  P 
ii  T—   i  P 

1.5628 
I.5643 
I-5654 

.0684 
.0616 
•0559 

1.0684 
1.0616 
L0559 

144 


BROWN    &    SHARPE    MFG.   CO. 


DIAMETRAL   PITCH. 

"NUTTALL." 
Diametral  Pitch  is  the  Number  of  Teeth  to  Each  Inch  of  the  Pitch  Diameter. 


To  Get 

Having 

Rule. 

Formula. 

The  Diametral 

3.1416 

Pitch. 

The  Diametral 
Pitch. 

The  Pitch  Diameter 
and  the  Number  of 

Divide  Number  of  Teeth  by  Pitch 

P' 

r=^ 

The  Diametral 
Pitch. 

Teeth  

The  Outside  Diame- 
ter and  the  Number 

Divide  Number  of  Teeth  plus  2  by 
Outside  Diameter    

P=^±l 

Pitch 
Diameter. 

of  Teeth  .... 

The  Number  of  Teeth 
and  the  Diametral 

Divide  Number  of  Teeth  by  the 
Diametral  Pitch                    . 

»'=T 

Pitch 

Pitch   
The  Numb.er  of  Teeth 

Divide    the   product    of    Outside 

!>•       D  X 

Diameter. 

eter      

bv  Number  of  Teeth  plus  2 

N+8 

Pitch 
Diameter. 

Pitch 
Diameter. 

The  Outside  Diame- 
ter and  the  Diam- 
etral Pitch  .    .    . 

Addendum    and   the 
Number  of  Teeth 

Subtract  from  the  Outside  Diame- 
ter the  quotient  of  2  divided  bv 
the  Diametral  Pitch     .... 

Multiply  Addendum  by  the  Num- 
ber of  Teeth        .    .  *  . 

!>•=.>-!- 

D'=sN 

Outside 

The  Number  of  Teeth 

Divide  Number  of  Teeth  plus  2  by 

D        N+2 

Diameter. 

Outside 
Diameter. 

Pitch  

The  Pitch  Diameter 
and  the  Diametral 
Pitch 

the  Diametral  Pitch    .... 

Add  to  the  Pitch  Diameter  the 
Siotient   of   2    divided    by  the 
iametral  Pitch                .    . 

P 

D  =  D'+Jr 

Outside 
Diameter. 

The  Pitch  Diameter 
and  the  Number  of 
Teeth  

Divide  the  Number  of  Teeth  plus 
2  bv  the  quotient  of  Number  of 
Teeth  and  bv  the  Pitch  Diameter 

y+2 

D' 

Outside 
Diameter. 

Number  of 
Teeth. 

The  Number  of  Teeth 
and  Addendum  . 

The  Pitch  Diameter 
and  the  Diametral 

Multiply  the    Number   of   Teeth 
plus  2  by  Addendum  .... 

Multiply  Pitch  Diameter  by  the 
Diametral  Pitch                ... 

D  =  (N+2)  s 
NsrD'P 

Number  of 
Teeth. 

Thickness 

Pitch   

The  Outside  Diame- 
ter and  the  Diame- 
tral Pitch     .    .    . 

Multiplv  Outside  Diameter  by  the 
Diametral  Pitch  and  subtract  2. 

Divide   1  5708   bv  the    Diametral 

N  =  DP  —  2 

1.5706 

of  Tooth. 

The  Diametral  Pitch. 

pitch    

P 

Addendum. 

The  Diametral  Pitch. 

Divide  1  bv  the  Diametral  Pitch, 
D' 

8  =  ^ 

-   ,   f        ]-157 

Root. 

Working 
Depth. 

The  Diametral  Pitch. 
The  Diametral  Pitch. 

Divide  1.157  by  the  Diametral  Pitch 
Divide  2  by  the  Diametral  Pitch. 

p 

-4- 

D»+  f       2'157 

Whole  Depth. 
Clearance. 
Clearance. 

The  Diametral  Pitch. 
The  Diametral  Pitch. 
Thickness  of  Tooth. 

I  )i  vide  2.15  1  by  the  Diametral  Pitch 
Divide  .157  by  the  Diametral  Pitch 
Divide    Thickness    of    Tooth    at 

«-^ 

f_    t 
—    in 

PROVIDENCE,    R.    I. 


145 


CIRCULAR   PITCH. 

"NUTTALL." 

Circular  Pitch  is  the  Distance  from  the  Centre  of  One  Tooth  to  the  Centre  of  the 
Next  Tooth,  Measured  along  the  Pitch  Circle. 


To  Get 

Having 

Rule. 

Formula. 

The  Circular 
Pitch 

The  Diametral  Pitch. 

Divide  3.1416  by  the  Diametral 
Pitch 

p,_  3.1416 

The  Circular 

The  Pitch   Diameter 

Divide    Pitch    Diameter   bv  the 

P 
P'          D/ 

Pitch. 

Teeth  

Teeth  .  ...  .  . 

.3183  N 

The  Circular 

The  Outside  Diame- 

Divide  Outside  Diameter  by  the 

P'              D 

Pitch. 

of  Teeth  .    .    . 

Teeth  plus  2 

.3183  N+2 

Pitch 
Diameter. 

The  Number  of  Teeth 
and    the     Circular 
Pitch  

The  continued  product  of  the 
Number  of  Teeth,  the  Circular 
Pitch  and  .3183  

D'-NP'.3183 

Pitch 

The  Number  of  Teeth 

Divide  the  product  of  Number  of 

D'        ND 

Diameter. 

Pitch 
Diameter. 

ameter    .... 

The  Outside  Diame- 
ter and  the  Circular 
Pitch  .... 

Number  of  Teeth  plus  2  ... 

Subtract  from  the  Outside  Diame- 
ter the  product  of  the  Circular 
Pitch  and  6366  ... 

N+2 
D'=D—  (P'.6366) 

Pitch 
Diameter 

Addendum    and    the 
Number  of  Teeth. 

Multiply  the  Number  of  Teeth  by 
the  Addendum  

D'=  N  S 

Outside 
Diameter. 

The  Number  of  Teeth 
and     the    Circular 
Pitch  

The  continued  product  of  the 
Number  of  Teeth  plus  2,  the 
Circular  Pitch  and  3183 

D=(N+2)P'.318S 

Outside 
Diameter. 

The  Pitch  Diameter 
and     the     Circular 
Pitch 

Add  to  the  Pitch  Diameter  the 
product  of  the  Circular  Pitch 
and  6366 

D=D+(P'.6366) 

Outside 
Diameter 

The  Number  of  Teeth 
and  the  Addendum 

Multiply  Addendum  by  Number 
of  Teeth  plus  2  

D  =  s  (N+2) 

Number  of 

The  Pitch  Diameter 

Divide  the  product  of  Pitch  Diam- 

XT       D'  3.1416 

Teeth. 

and    the     Circular 
Pitch   .    .    „    .    . 

eter  and  3.1416  by  the  Circular 
Pitch  

P' 

Thickness 
of  Tooth. 

Addendum. 

The  Circular  Pitch. 
The  Circular  Pitch. 

One  -half  the  Circular  Pitch   .    . 

Multiply  the  Circular  Pitch  by 
.3183,  or  s  —  —  

P' 
=  ~2~ 

s  =  P'  .3183 

Root. 

The  Circular  Pitch. 

Multiply  the  Circular  Pitch  by 
.3683  

s  +  f  =  P'  .3683 

Working 
Depth 

The  Circular  Pitch.    , 

Multiply  the  Circular  Pitch  by 
6366 

D"=  P'  .6366 

Whole  Depth. 

The  Circular  Pitch. 

Multiply   the    Circular  Pitch   by 
(5866      

!)"=  P'  .6866 

Clearance. 
Clea  -ance. 

The  Circular  Pitch. 
Thickness  of  Tooth. 

Multiply  the  Circular  Pitch  by  .05 

One-tenth  the  Thickness  of  Tooth 
at  Pitch  Line  

f  =  P  .05 

f  =  TF 

146  BROWN   &    SHARPE   MFG.    CO. 

GEAR  WHEELS. 

TABLE   OF    TOOTH   PARTS CIRCULAR    PITCH   IN    FIRST    COLUMN. 


Circular 
Pitch. 

Threads  or 
Teeth  per  inch 
Linear  . 

Diametral 
Pitch. 

Thickness  of 
Tooth  on 
Pitch  Line. 

Addendum 
and  Module. 

<. 
Working  Depth 

of  Tooth. 

CD 

c3        o 

*Bl 

J2J 

$  a 
« 

Whole  Depth 
of  Tooth. 

-L 

A-L  £ 

£lH 

,p 

,1 

5  -8 

^    n3 

£     1 
1 

P' 

7' 

P 

t 

s 

D" 

•+/ 

D"+/ 

P'X.31 

P'X.335 

2 

i 

2 

1.5708 

1.0000 

.6366 

1.2732 

.7366 

1.3732 

.6200 

.6700 

If 

8 
15 

1.6755 

.9375 

.5968 

1.1937 

.6906 

1.2874 

.5813 

.6281 

H 

i 

1.7952 

.8750 

.5570 

1.1141 

.6445 

1.2016 

.5425 

.5863 

H 

8 
13" 

1.9333 

.8125 

.5173 

1.0345 

.5985 

1.1158 

.5038 

.5444 

H 

2 
3 

2.0944 

.7500 

.4775 

.9549 

.5525 

1.0299 

.4650 

.5025 

ifc 

16 
23 

2.1855 

.7187 

.4576 

.9151 

.5294 

.9870 

.4456 

.4816 

11 

8 
11 

2.2848 

.6875 

.4377 

.8754 

.5064 

.9441 

.4262 

.4606 

l± 

3 
4 

2.3562 

.6666 

.4244 

.8488 

.4910 

.9154 

,4133 

.4466 

i* 

16 
21 

2.3936 

.6562 

4178 

.8356 

.4834 

.9012 

.4069 

.4397 

H 

4 
5 

2.5133 

.6250 

.3979 

.7958 

.4604 

.8583 

.3875 

.4188 

1  3 

-I  16 

16 
19" 

2.6456 

.5937 

.3780 

.7560 

.4374 

.8156 

.3681 

.3978 

1± 

_L 
9 

2.7925 

.5625 

.3581 

.7162 

.4143 

.7724 

.3488 

.3769 

1* 

f 

2.9568 

.5312 

.3382 

.6764 

.3913 

.7295 

.3294 

.3559 

1 

1 

3.1416 

.5000 

3183 

.6366 

.3683 

.6866 

.3100 

.3350 

J» 

16 

1* 

3.3510 

.4687 

2984 

.5968 

.3453 

.6437 

.2906 

.3141 

7 
8 

11 

3.5904 

.4375 

2785 

.5570 

.3223 

.6007 

.2713 

.2931 

# 

1* 

3.8666 

.4062 

2586 

.5173 

.2993 

.5579 

.2519 

.2722 

f 

11 

3.9270 

.4000 

2546 

.5092 

.2946 

.5492 

.2480 

.2680 

J_ 
4 

11 

4.1888 

.3750 

2387 

.4775 

2762 

.5150 

.2325 

.2513 

# 

1* 

4.5696 

.3437 

.2189 

.4377 

.2532 

.4720 

.2131 

.2303 

2 
3 

11 

4.7124 

.3333 

2122 

.4244 

2455 

.4577 

2066 

.2233 

5 

11 

5.0265 

.3125 

1989 

.3979 

.2301 

.4291 

.1938 

.2094 

J_ 

11 

5.2360 

.3000 

1910 

.3820 

.2210 

.4120 

1860 

.2010 

± 

11 

5.4978 

.2857 

1819 

.3638 

.2105 

.3923 

1771 

1914 

1 

11 

5.5851 

.2812 

1790 

.3581 

2071 

.3862 

1744 

.1884 

To  obtain  the  size  of  any  part  of  a  circular  pitch  not  given  -n    the 
table,     multiply     the     corresponding     part     1"     pitch     by     the     pitch 
required. 

PROVIDENCE,    R.    I. 
TABLE  OF  TOOTH  PARTS. — Continued. 

CIRCULAR    PITCH    I2T    FIRST    COLUMN. 


147 


Circular 
Pitch. 

Threads  or  _ 
Teeth  per  inch 
Linear. 

Diametral 
Pitch. 

TJlickness  of 
Tooth  on 
Pitch  Line. 

Addendum 
and  Module. 

I 

<D  _.J 

fi| 

&D  0 

.2  ^ 
-2*3 

o 

£ 

1*1 

•8-2^ 

0  0>  rj 

*A| 

$    fl 

rCj 
-+J 

*i 

I* 

ill 

slw 

£1* 

H 

~J 
a* 

HH  ^ 
K^  o3 

ii 

P' 

£ 

P 

t 

s  y 

D" 

•+/ 

Tfrf. 

Pk.3i 

PX.335 

i 

2 

2 

6.2832 

.2500 

.1592 

.3183 

.1842 

.3433 

.1550 

.1675 

4 

T 

21 

7.0685 

.2222 

.1415 

.2830 

.1637 

.3052 

.1378 

.1489 

7 
16 

2f 

7.1808 

.2187 

.1393 

.2785 

.1611 

.3003 

.1356 

.1466 

3 

7 

2f 

7.3304 

.2143 

.1364 

.2728 

.1578 

.2942 

.1328 

.1436 

j^ 
5 

21- 

7.8540 

.2000 

.1273 

.2546 

.1473 

.2746 

.1240 

.1340 

3 
8 

0.2. 
Z3 

8.3776 

.1875 

.1194 

.2387 

.1381 

.2575 

.1163 

.1256 

i 
11 

0.1 

^4 

8.6394 

.1818 

.1158 

.2313 

.1340 

.2498 

.1127 

.1218 

1 
T 

3 

9.4248 

.1666 

.1061 

.2122 

.1228 

.2289 

.1033 

.1117 

5 
16 

3i 

10.0531 

.1562 

.0995 

.1989 

.1151 

.2146 

.0969 

.1047 

.J_ 
10 

3± 

10.4719 

.1500 

.0955 

.1910 

.1105 

.2060 

.0930 

.1005 

2 

7 

3i 

10.9956 

.1429 

.0909 

.1819 

.1052 

.1962 

.0886 

.0957 

- 

4 

12.5664 

.1250 

.0796 

.1591 

.0921 

.1716 

.0775 

.0838 

— 

41 

14.1372 

.1111 

.0707 

.1415 

.0818 

.1526 

.0689 

.0744 

— 

5 

15.7080 

.1000 

.0637 

.1273 

.0737 

.1373 

.0620 

.0670 

3 

10 

5i 

16.7552 

.0937 

.0597 

.1194 

.0690 

.1287 

.0581 

.0628 

2 
11 

5f 

17.2788 

.0909 

.0579 

.1158 

.0670 

.1249 

.0564 

.0609 

1 

T 

6 

18.8496 

.0833 

.0531 

.1061 

.0614 

.1144 

.0517 

.0558 

2 
13 

6i 

20.4203 

.0769 

.0489 

.0978 

.0566 

.1055 

.0477 

.0515 

1 

7 

7 

21.9911 

.0714 

.0455 

.0910 

.0526 

.0981 

.0443 

.0479 

2 
15 

7i 

23.5619 

.0666 

.0425 

.0850 

.0492 

.0917 

.0414 

.0446 

1 

8 

8 

25.1327 

.0625 

.0398 

.0796 

.0460 

.0858 

.0388 

.0419 

1 

9 

9 

28.2743 

.0555 

.0354 

.0707 

.0409 

.0763 

.0344 

.0372 

JL 
10 

10 

31.4159 

.0500 

.0318 

.0637 

.0368 

.0687 

.0310 

.0335 

1 

1C 

16 

50.2655 

.0312 

.0199 

.0398 

.0230 

.0429 

.0194 

.0209 

W 

20 

62.8318 

.0250 

.0159 

.0318 

.0184 

.0343 

.0155 

.0167 

To  obtain  the 

table,     multiply 
required. 


size  of  any  part  or 
the     corresponding 


a  circular  pitch  not  given  in    tl/e 
part     1"     pitch     by     the     pitch 


148 


BROWN    &    SHAEPE    XFG.    CO. 


GEAR  WHEELS. 


TABLE  OF  TOOTH  PARTS DIAMETRAL    PITCH   IN    FIRST    COLUMN. 


Diametral 
Pitch. 

f| 

Thickness 
of  Tooth  on 
Pitch  Line. 

Addendum 

and  1^-' 

Working  Depth 
of  Tooth. 

I  s 

"8.!^ 

Q 

jP 

P 

P' 

t 

s 

D" 

»+/. 

D"+/. 

* 

6.2832 

3.1416 

2.0000 

4.0000 

2.3142 

4.3142 

f 

4.1888 

2.0944 

1.3333 

2.6666 

1.5428 

2.8761 

1 

3.1416 

1.5708 

1.0000 

2.0000 

1.1571 

2.1571 

1J 

2.5133 

1.2566 

.8000 

1.6000 

.9257 

1.7257 

1J 

2.0944 

1.0472 

.6666 

1.3333 

.7714 

1.4381 

If 

1.7952 

.8976 

.5714 

1.1429 

.6612 

1.2326 

2 

1.5708 

.7854 

.5000 

1.0000 

.5785 

1.0785 

at 

1.3963 

.6981 

.4444 

.8888 

.5143 

.9587 

2£ 

1.2566 

.6283 

.4000 

.8000 

.4628 

.8628 

2f 

1.1424 

.5712 

.3636 

.7273 

.4208 

.7844 

3 

1.0472 

.5236 

.3333 

.6666 

.3857 

.7190 

3J 

.8976 

.4488 

.2857 

.5714 

.3306 

.6163 

4 

.7854 

.3927 

.2500 

.5000 

.2893 

.5393 

5 

.6283 

.3142 

.2000 

.4000 

.2314 

.4314 

6 

.5236 

.2618 

.1666 

.3333 

.1928 

.3595 

7 

.4488 

.2244 

.1429 

.2857 

.1653 

.3081 

8 

.3927 

.1963 

.1250 

.2500 

.1446 

.2696 

9 

.3491 

.1745 

.1111 

.2222 

.1286 

.2397 

10 

.3142 

.1571 

.1000 

.2000 

.1157 

.2157 

11 

.2856 

.1428 

.0909 

.1818 

.1052 

.1961 

12 

.2618 

.1309 

0833 

.1666 

.0964 

.1798 

13 

.2417 

.1208 

.0769 

.1538 

.0890 

.1659 

14 

.2244 

.1122 

.0714 

.1429 

.0826 

.1541 

To  obtain  the  size  of  any  part  of  a  diametral  pitch  not  given  in  the 
table,  divide  the  correspond  iri"  part  of  1  diametral  pitch  by  the  pitch 
required. 


PROVIDENCE,  R.   I. 


149 


TABLE  OF  TOOTH  PAETS—  Continued. 

DIAMETRAL   PITCH   IN   FIRST    COLUMN. 


Diametral 
Pitch. 

b 

S3  « 
11 

CffJ 

b 

Thickness 
of  Tooth  on 
Pitch  Line. 

Addendum 
and  jp 

1 
|| 

p 

u 

00  £-2 
1  S 

Id 

P. 

P'. 

t. 

s. 

D". 

s+f. 

P"-*./ 

15 

.2094 

.1047 

.0666 

.1333 

.0771 

.1438 

16 

.1963 

.0982 

.0625 

.1250 

.0723 

.1348 

17 

.1848 

.0924 

.05-8 

.1176 

.0681 

.1269 

18 

.1745 

.0873 

.0555 

.1111 

.0643 

.1198 

19 

.1653 

.0827 

.0526 

.1053 

.0609 

.1135 

20 

.1571 

.0785 

.0500 

.1000 

.0579 

.1079 

22 

.1428 

.0714 

.0455 

.0909 

.0526 

.0980 

24 

.1309 

.0654 

.0417 

.0833 

.0482 

.0898 

26 

.1208 

.0604 

.0385 

.0769 

.0445 

.0829 

28 

.1122 

.0561 

.0357 

.0714 

.0413 

.0770 

30 

.1047 

.0524 

.0333 

.0666 

.0386 

.0719 

32 

.0982 

.0491 

.0312 

.0625 

.0362 

.0674 

34 

.0924 

.0462 

.0294 

.0588 

.0340 

.0634 

36 

.0873 

.0436 

.0278 

.0555 

.0321 

.0599 

38 

.0827 

.0413 

.0263 

.0526 

.0304 

.0568 

40 

.0785 

.0393 

.0250 

.0500 

.0289 

.0539 

42 

.0748 

.0374 

.0238 

.0476 

.0275 

.0514 

44 

.0714 

.0357 

.0227 

.0455 

.0263 

.0490 

46 

.0683 

.0341 

.0217 

.0435 

.0252 

.0469 

48 

.0654 

.0327 

.0208 

.0417 

.0241 

.0449 

50 

.0628 

.0314 

.0200 

.0400 

.0231 

.0431 

56 

.0561 

.0280 

.0178 

.0357 

.0207 

.0385 

60 

.0524 

.0262 

.0166 

.0333 

.0193 

.  0360 

To  obtain  the  size  of  any  part  of  a  diametral  pitch  not  given  in  the 
table,  divide  the  corresponding  part  of  1  diametral  pitch  by  the  pitch 
.required. 


150 


BP.OWN    k    SHAKPE    MFG.    CO. 


NATURAL   SINE. 


Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

0 

.00000 

.00291 

.00581 

.00872 

.01103 

.01454 

.01745 

89 

1 

.01745 

.02036 

.02326 

.02617 

.02908 

.03199 

.03489 

88 

2 

.03489 

.03780 

.04071 

.04361 

.04652 

.04943 

.05233 

87 

3 

.05203 

.05524 

.05814 

.06104 

.06395 

.06685 

.06975 

8G 

4 

.06975 

.07265 

.07555 

.07845 

.08135 

.08425 

.08715 

85 

5 

.08715 

.09005 

.09295 

.09584 

.09874 

.10163 

.10452 

84 

6 

.10452 

.10742 

.11031 

.11320 

.11609 

.11898 

.12186 

80 

7 

.12186 

.12475 

.12764 

.13052 

.  13341 

.13629 

.13917 

82 

8 

.13917 

.14205 

.14493 

.14780 

.15068 

.15356 

.15643 

81 

9 

.15643 

.15980 

.16217 

.16504 

.16791 

.17078 

.17364 

80 

10 

.  17364 

.17651 

.17937 

.18223 

.  18509 

.18795 

.19080 

79 

11 

.19080 

.19366 

.19651 

.19936 

.20221 

.20506 

.20791 

78 

12 

.20791 

.21075 

.21359 

.21644 

.21927 

.22211 

.22495 

77 

13 

.22495 

.22778 

.23061 

.23344 

.23627 

.23909 

.24192 

76 

14 

.24192 

.24474 

.24756 

.25038 

.25319 

.25600 

.25881 

75 

15 

.25881 

.261C2 

.26443 

.26723 

.27004 

.27284 

.27563 

74 

16 

.27563 

.27843 

.28122 

.28401 

.28680 

.28958 

.29237 

73 

17 

.29237 

.29515 

.29793 

.30070 

.30347 

.30624 

.30901 

72 

18 

.30901 

.31178 

.31454 

.31730 

.32006 

.32281 

.32556 

71 

19 

.32556 

.32831 

.33106 

.33380 

.33654 

.33928 

.34202 

70 

20 

.34202 

.34475 

.34748 

.35020 

.35298 

.35565 

.35836 

69 

21 

.35836 

.36108 

.36379 

.36650 

.36920 

.37190 

.37460 

68 

22 

.37460 

.377GO 

.37999 

.38268 

.38536 

.38805 

.39073 

67 

23 

.39073 

.39340 

.39607 

.89874 

.4Qfctl 

.40407 

.40673 

66 

24 

.40673 

.409C9 

.41204 

.41469 

.41733 

.41998 

.42261 

65 

25 

.42261 

.42525 

.42788 

.43051 

.43313 

.43575 

.43887 

64 

26 

.43837 

.44098 

.44359 

.44619 

.44879 

.45139 

.45399 

63 

27 

.45399 

.45658 

.45916 

.46174 

.46432 

.46690 

.46947 

62 

28 

.46947 

.47203 

.47460 

.47715 

.47971 

.48226 

.48481 

61 

29 

.48481 

.48735 

.48989 

.49242 

.49495 

.49747 

.50000 

60 

30 

.50000 

.50251 

.50503 

.50753 

.51004 

.51254 

.51503 

59 

31 

.51503 

.51752 

.52001 

.52249 

.52497 

.52745 

.52991 

1  58 

32 

.52991 

.532C8 

.53484 

.53730 

.53975 

.54219 

.54463 

57 

33 

.54463 

.54707 

.54950 

.55193 

.55436 

.55677 

.55919 

56 

34 

.55919 

.56160 

.56400 

.56640 

.56880 

.57119 

.57357 

55 

35 

.57357 

.57595 

.57833 

.58070 

.58306 

.58542 

.58778 

54 

36 

.58778 

.59013 

.59248 

.59482 

.59715 

.59948 

.60181 

58 

37 

.60181 

.60413 

.60645 

.60876 

.61106 

.61336 

.61566 

5-3 

38 

.61566 

.61795 

.62023 

.02251 

.62478 

.62705 

.62932 

51 

39 

.62932 

.63157 

.63383 

.63607 

.63832 

.64055 

.64278 

50 

40 

.64278 

.64501 

.64723 

.64944 

.65165 

.65386 

.65605 

49 

41 

.65605 

.65825 

.66043 

.66262 

.66479 

.66696 

.66913 

48 

42 

.66913 

.67128 

.67344 

.67559 

.67773 

.67986 

.68199 

47 

43 

.68199 

.68412 

.68624 

.68835 

.69046 

.69256 

.69465 

4G 

44 

.69465 

.69674 

.69883 

.  70090 

.70298 

.70504 

.70710 

45 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

i>eg. 

NATURAL   COSINE. 


PROVIDENCE,    R.    I. 


151 


NATURAL   SINE. 


Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

&y 

45 

.70710 

.70916 

.71120 

.71325 

.71528 

.71731 

.71934 

44 

46 

.71934 

.72135 

.72336 

.72537 

.72737 

.72936 

.73135 

43 

47 

.73135 

.73333 

.  73530 

.73727 

.73923 

.74119 

.74314 

42 

48 

.74314 

.74508 

.74702 

.74895 

.75088 

.75279 

.75471 

41 

49 

.75471 

.75661 

.75851 

.76040 

.76229 

.76417 

.76604 

40 

50 

.76G04 

.76791 

.76977 

.771G2 

.77347 

.77531 

.77714 

39 

51 

.77714 

.77897 

.78079 

.78260 

.78441 

.78621 

.  78801 

38 

53 

.78801 

.78979 

.79157 

.  793S5 

.79512 

.79683 

.79863 

37 

53 

.79863 

.80038 

.80212 

.  80385 

.80558 

.80730 

.80901 

36 

54 

.80901 

.81072 

.81242 

.81411 

.81580 

.81748 

.81915 

35 

55 

[  .81915 

.82081 

.82247 

.82412 

.82577 

.82740 

.82903 

84 

56 

.82903 

.83066 

.83227 

.83383 

.83548 

.83708 

.83867 

33 

57 

.83867 

.84025 

.84182 

.84339 

.84495 

.84650 

.84804 

32 

53 

.84804 

.84958 

.85111 

.85264 

.85415 

.85566 

.85716 

31 

59 

.85716 

.85866 

.86014 

.86162 

.86310 

.86456 

.86602 

30 

60 

.86602 

.86747 

.86892 

.87035 

.87178 

.87320 

.87462 

29 

61 

.87462 

.87602 

.87742 

.87881 

.88020 

.88157 

.88294 

28 

62 

.88294 

.88430 

.88566 

.88701 

.88835 

.88968 

.89100 

27 

C3 

.89fOO 

.89232 

.89363 

.89493 

.89622 

.89751 

.89879 

26 

64 

.89870 

.90006 

.90132 

.90258 

.90383 

.90507 

.90630 

25 

65 

.90630 

.90753 

.90875 

.90996 

.91116 

.91235 

.91354 

24 

66 

.91354 

.91472 

.91589 

.91706 

.91821 

.91936 

.92050 

23 

67 

.92050 

.92163 

.92276 

.92388 

.92498 

.92609 

.92718 

22 

C8 

.92718 

.92827 

.92934 

.93041 

.93148 

.93253 

.93358 

21 

69 

.93358 

.93461 

.93565 

.93667 

.93768 

.93869 

.93969 

20 

70 

.93969 

.94068 

.94166 

.94264 

.94360 

.94456 

.94551 

19 

71 

.94551 

.94646 

.94789 

.94832 

.94924 

.95015 

.95105 

18 

72 

.95105 

.95195 

.95283 

.95371 

.95458 

.95545 

.95630 

17 

73 

.95630 

.95715 

.95799 

.95882 

.95964 

.96045 

.96126 

16 

74 

.96126 

.96205 

.96284 

.96363 

.96440 

.96516 

.96592 

15 

75 

.96592 

.96667 

.96741 

.96814 

.96887 

.96958 

.97029 

14 

76 

.97029 

.97099 

.97168 

.97237 

.97304 

.97371 

.97437 

13 

77 

.97437 

.97502 

.97566 

.97629 

.97692 

.97753 

.97814 

12 

78 

.97814 

.97874 

.97934 

.97992 

.98050 

.98106 

.98162 

11 

79 

.98162 

.98217 

.98272 

.98325 

.98378 

.98429 

.98480 

10 

80 

.98480 

.98530 

.98580 

.98628 

.98676 

.98722 

.98768 

9 

81 

.98768 

.98813 

.98858 

.98901 

.98944 

.98985 

.99026 

8 

82 

.99026 

.99066 

.99106 

.99144 

.99182 

.99218 

.99254 

7 

83 

.99254 

.99289 

.99323 

.99357 

.99389 

.99421 

.99452 

6 

84 

.99452 

.99482 

.99511 

.99539 

.99567 

.99593 

.99619 

5 

85 

.99619 

.99644 

.99668 

.99691 

.99714 

.99735 

.99756 

4 

86 

.99756 

.99776 

.99795 

99813 

.99830 

.99847 

.99863 

3 

87 

.99863 

.99877 

.99891 

.99904 

.99917 

.99928 

.99939 

2 

.  88 

.99939 

.99948 

.99957 

.99965 

.99972 

.99979 

.99984 

1 

89 

.99984 

.99989 

.99993 

.99996 

.99998 

.99999 

1.0000 

0 

- 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATURAL   COSINE. 


152 


BROWN    &    SHAEPE    MFG.    CO. 


NATURAL  TANGENT. 


Deg. 

(X 

10' 

20' 

30' 

40' 

50' 

60' 

0 

.00000 

.00290 

.00581 

.00872 

.01163 

.01454 

.01745 

89 

1 

.01745 

.02036 

.02327 

.02618 

.02909 

.03200 

.03492 

88 

8 

.03492 

.03783 

.04074 

.04366 

.04657 

.04949 

.05240 

87 

3 

.05240 

.05532 

.05824 

.06116 

.06408 

.06700 

.06992 

86 

4 

.06992 

.07285 

.07577 

.07870 

.08162 

.08455 

.08748 

85 

5 

.08748 

.09042 

.09335 

.09628 

.09922 

.10216 

.10510 

84 

G 

.10510 

.10804 

.11099 

.11393 

.11688 

.11983 

.12278 

83 

7 

.12278 

.12573 

.12869 

.13165 

.13461 

.  13757 

.14054 

82 

8 

.14054 

.14350 

.14647 

.14945 

.15242 

.15540 

.15838 

81 

9 

.  15838 

.16136 

.16435 

.16734 

.17033 

.17332 

.17632 

80 

10 

.17632 

.  17932 

.18233 

.18533 

.18834 

.19136 

.19438 

79 

11 

.19438 

.19740 

.20042 

.20345 

.20648 

.20951 

.21255 

78 

12 

.21255 

.21559 

.21864 

.22169 

.22474 

.22780 

.23086 

77 

13 

.23086 

.23393 

.23700 

.24007 

.24315 

.24624 

.24932 

76 

14 

.24932 

.25242 

.25551 

.25861 

.26172 

.26483 

.26794 

75 

15 

.26794 

.27106 

.27419 

.27732 

.28046 

.28360 

.28674 

74 

16 

.28674 

.28989 

.29305 

.29621 

.29938 

.30255 

.30573 

73 

17 

.30573 

.30891 

.31210 

.31529 

.31850 

.32170 

.32492 

72 

13 

.32492 

.32813 

.33136 

.33459 

.33783 

.34107 

.34432 

71 

19 

.34482 

.34758 

.35084 

.35411 

.35739 

.36067 

.86397 

70 

20 

.36397 

.36726 

.37057 

.37388 

.37720 

.38053 

.38386 

69 

21 

.38386 

.38720 

.39055 

.39391 

.39727 

.40064 

.40402 

68 

22 

.40402 

.40741 

.41080 

.41421 

.41762 

.42104 

.42447 

67 

23 

.42447 

.42791 

.43135 

.43481 

.43827 

.44174 

.44522 

60 

24 

.44522 

.44871 

.45221 

.45572 

.45924 

.46277 

.46630 

65 

25 

.46630 

.46985 

.47341 

.47697 

.48055 

.48413 

.48773 

64 

26 

.48773 

.49133 

.49495 

.49858 

.50221 

.50586 

.50952 

63 

27 

.50952 

.51319 

.51687 

.52056 

.52427 

.52798 

.53170 

62 

28  | 

.53170 

.53544 

.53919 

.54295 

.54672 

.55051 

.55430 

61 

29  ' 

.55430 

.55811 

.56193 

.56577 

.56961 

.57847 

.57735 

60 

30  I 

.57735 

.58123 

.58513 

.58904 

.59297 

.59690 

.60086 

£9 

31 

.60086 

.60482 

.60880 

.61280 

.61680 

.62083 

.62486 

58 

32 

.62486 

.62892 

.63298 

.63707 

.64116 

.64528 

.64940 

57 

33 

.64940 

.65355 

.65771 

.66188 

.66607 

.67028 

.67450 

56 

34 

.67450 

.67874 

.68300 

.68728 

.69157 

.69588 

.70020 

55 

35 

.70020 

.70455 

.70891 

.71329 

.71769 

.72210 

.72654 

54 

36 

.72654 

.73099 

.73546 

.73996 

.74447 

.74900 

.75355 

53 

37 

.75355 

.75812 

.76271 

.76732 

.77195 

.77661 

.78128 

52 

38 

.78128 

.78598 

.79069 

.79543 

.80019 

.80497 

,  80978 

51 

39 

.80978 

.81461 

.81946 

.82433 

.82923 

.83415 

.83910 

50 

40 

.83910 

.84406 

.84906 

.85408 

.85912 

.86419 

.86928 

49 

41 

.86928 

.87440 

.87955 

.88472 

.88992 

.89515 

.90040 

48 

42 

.90040 

.90568 

.91099 

.91633 

.92169 

.92709 

.93251 

47 

43 

.93251 

.93796 

.94345 

.94896 

.95450 

.96008 

.96568 

46 

44 

.96568 

.97132 

.97699 

.98269 

.98843 

.99419 

1.0000 

45 

a 

607 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATURAL  COTANGENT. 


PROVIDENCE,  R.  I. 


153 


NATUBAL  TANGENT. 


Deg. 

o' 

10' 

20' 

30' 

40' 

50' 

60 

45 

1.0000 

1.0058 

1.0117 

1.0176 

1.0235 

1.0295 

1.0355 

44 

46 

1.0355 

1.0415 

1.0476 

1.0537 

1.0599 

1.0661 

1.0723 

43 

47 

1.0723 

1.0786 

.0849 

1.0913 

1.0977 

1.1041 

1.1106 

42 

48 

1.1106 

1.1171 

.1236 

1  .  1302 

1.1369 

1.1436 

1.1503 

41 

49 

1.1503 

1.1571 

1639 

1.1708 

1.1777 

1.1847 

1.1917 

40 

50 

1.1917 

1.1988 

.2059 

1.2131 

1.2203 

1.2275 

1  2349 

39 

51 

1.2349 

1.2422 

.2496 

1.2571 

1.2647 

1.2723 

1.2799 

38 

52 

1.2799 

1.2876 

.2954 

1.3032 

1.3111 

1.3190 

1.3270 

37 

53 

1.3270 

1.3351 

.3432 

1.3514 

1.3596 

1.3680 

1.3763 

36 

54 

1.3763 

1.3848 

.3933 

1.4019 

1.4106 

1.4193 

1.4281 

35 

55 

1.4281 

1.4370 

.4459 

1.4550 

1.4641 

1.4733 

1.4825 

34 

56 

1  4825 

1.4919 

1.5013 

1.5108 

1.5204 

1.5301 

1.5398 

33 

57 

1.5398 

1.5497 

1.5596 

1.5696 

1.5798 

1.5900 

1.6003 

32 

58 

1.6003 

1.6107 

1.6212 

1.6318 

1.6425 

1.6533 

1.6642 

31 

59 

1.6642 

1.6753 

1.6864 

1.6976 

1.7090 

1.7204 

1.7320 

30 

60 

1.7320 

1.7437 

1.7555 

1.7674 

1.7795 

1.7917 

1.8040 

29 

61 

1.8040 

1.8164 

1.8290 

1.8417 

1.8546 

1.8676 

1.8807 

28 

62 

1.8807 

1.8940 

1.9074 

1.9209 

1.9347 

1.9485 

1.9626 

27 

63 

1.9626 

1.9768 

1.9911 

2.0056 

2.0203 

2.0352 

2.0503 

26 

64 

2.0503 

2.0655 

2.0809 

2.0965 

2.1123 

2.1283 

2.1445 

25 

65 

2.1445 

2.1609 

2.1774 

2.1943 

2.2113 

2.2285 

2.2460 

24 

66 

2.2460 

2.2637 

2.2816 

2.2998 

2.3182 

2.3369 

2.3558 

23 

67 

2.3558 

2.3750 

2.3944 

2.4142 

2.4342 

2.4545 

2.4750 

22 

68 

2.4750 

2.4959 

2.5171 

2.5386 

2.5604 

2.5826 

2.6050 

21 

69 

2.6050 

2.6279 

2.6510 

2.6746 

2.6985 

2.7228 

2.7474 

20 

70 

2.7474 

2.7725 

2.7980 

2.8239 

2.8502 

2.8770 

2.9042 

19 

71 

2.9042 

2.9318 

2.9600 

2.9886 

3.0178 

3.0474 

3.0776 

18 

72 

3.0776 

3.1084 

3.1397 

3.1715 

3.2040 

3.2371 

3.2708 

17 

73 

3.2708 

3.3052 

3.3402 

3.3759 

3.4123 

3.4495 

3.4874 

16 

74 

3.4874 

3.5260 

3.5655 

3.6058 

3.6470 

3.6890 

3.7320 

15 

75 

3.7320 

3.7759 

3.8208 

3.8667 

3.9136 

3.9616 

4.0107 

14 

76 

4.0107 

4.0610 

4.1125 

4.1653 

4.2193 

4.2747 

4.3314 

13 

77 

4.  3314 

4.3896 

4.4494 

4.5107 

4.5736 

4.6382 

4.7046 

12 

78 

4.7046 

4.7728 

4.8430 

4.9151 

4.9894 

5.0658 

5.1445 

11 

79 

5.1445 

5.2256 

5.3092 

5.3955 

5.4845 

5.5763 

5.6712 

10 

80 

5.6712' 

5.7693 

5.8708 

5.9757 

6.0844 

6.1970 

6.3137 

9 

81 

6.3137 

6.4348 

6.5605 

6.6911 

6.8269 

6.9682 

7.1153 

8 

82 

7.1153 

7.2687 

7.4287 

7.5957 

7.7703 

7.9530 

8.1443 

7 

83 

S.1443 

8.3449 

8.5555 

8.7768 

9.0098 

9.2553 

9.5143 

6 

84 

9.5143 

9.7881 

10.078 

10.385 

10.711 

11.059 

11.430 

5 

85 

11.430 

11.826 

12.250 

12.706 

13.196 

13.726 

14.300 

4 

86 

14.300 

14.924 

15.604 

16.349 

17.169 

18.075 

19.081 

3 

87 

19.081 

20.205 

21.470 

22.904 

24.541 

26.431 

28.636 

2 

88 

28.636 

31.241 

34.367 

38.188 

42.964 

49.103 

57.290 

1 

89 

57.290 

68.750 

85.939 

114.58 

171.88 

343.77 

00 

0 

60' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATURAL  COTANGENT. 


154 


BEOWN    &    SHAEPE    MFG.    CO. 


NATURAL   SECANT. 


Deg. 

0' 

10' 

20' 

30' 

43' 

50 

60' 

0 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0001 

1.0001 

89 

1 

l.OOOL 

1.0002 

1.0002 

1.0003 

1.0004 

1.0005 

1.000(3 

88 

2 

1.0006 

1.0007 

1.0008 

1.0009 

1.0010 

1.0012 

1.0013 

87 

3 

1.0013 

1.0015 

1.0016 

1.0018 

1.0020 

1.0022 

1.0024 

86 

4 

1.0024 

1.0026 

1.0028 

1.0030 

1.0033 

1.0035 

1.0038 

85 

5 

1.0038 

1.0040 

1.0043 

1.0046 

1.0049 

1.0052 

1.0055 

84 

6 

1.0055 

1.0058 

1.0061 

1.0064 

1.0068 

1.0071 

1.0075 

83 

7 

1.0075 

1.0078 

1.0082 

1.0086 

1.0090 

1.0094 

1.0098 

82 

8 

1.0098 

1.0102 

1.0106 

1.0111 

1.0115 

1.0120 

1.0124 

81 

9 

1.0124 

1.0129 

1.0134 

1.0139 

1.0144 

1.0149 

1.0154 

80 

10 

1.0154 

1.0159 

1.0164 

1.0170 

1.0175 

1.0181 

1.0187 

79 

11 

1.0187 

1.0192 

1.0198 

1.0204 

1.0210 

1.0217 

1.0223 

78 

12 

1.0223 

1.0229 

1.0236 

1.0242 

1.0249 

1.0256 

1.0263 

77 

13 

1.0263 

1.0269 

1.0277 

1.0284 

1.0291 

1.0298 

1.0308 

76 

14 

1.0308 

1.0313 

1.0321 

1.0329 

1.0336 

1.0344 

1.0352 

75 

15 

1.0352 

1.0360 

1.0369 

1.0377 

1.0385 

1.0394 

1.0402 

74 

16 

1.0403 

1.0411 

1.0420 

1.0429 

1.0438 

1.0447 

1.0456 

73 

17 

1.0456 

1.0466 

1.0475 

1.0485 

1.0494 

1.0504 

1.0514 

72 

18 

1.0514 

1.0524 

1.0534 

1.0544 

1.0555 

1.0565 

1.0576 

71 

19 

1.0576 

1.0586 

1.0597 

l.Ofa'08 

1.0619 

1  0630 

1.0641 

70 

20 

1.0641 

1.0653 

1.0664 

1  .  0676 

1.0087 

1.0699 

1.0711 

69 

21 

1.0711 

1.0723 

1.0735 

1.0747 

1.0760 

1.0772 

1.0785 

68 

22 

1.0785 

1.0798 

1.0810 

1.0823 

1.0837 

1.0850 

1.0863 

67 

23 

1.0863 

1.0877 

1.0890 

1.0904 

1.0918 

1.0932 

1.0946 

66 

24 

1.0946 

1.0960 

1.0974 

1.0989 

1.1004 

1.1018 

1.1033 

65 

25 

1.1033 

1.1048 

1.1063 

1.1079 

1.1094 

1.1110 

1.1126 

64 

26 

1.1126 

1.1141 

1.1157 

1.1174 

1.1190 

1.1206 

1.1223 

63 

27 

1.1223 

1.1239 

1.1256 

1.1273 

1.1290 

1.1308 

1.1325 

62 

28 

1.1325 

1  .  1343 

1.1361 

1.1378 

1.1396 

1.1415 

1.1433 

61 

29 

1.1433 

1.1452 

1.1470 

1.1489 

1.1508 

1  .  1527 

1.1547 

60 

30 

1.1547 

1.1566 

1.1586 

1.1605 

1.1625 

1.1646 

1.1666 

59 

31 

1.1666 

1.1686 

1.1707 

1.1728 

1.1749 

1.1770 

1.1791 

58 

32 

1.1791 

1.1813 

1.1835 

1.1856 

1.1878 

1.1901 

1.1923 

57 

33 

1.1923 

1.1946 

1.1969 

1.1992 

1.2015 

1.2038 

1.2062 

56 

34 

1.2032 

1.2085 

1.2109 

1.2134 

1.2158 

1.2182 

1.2207 

55 

35 

1.2207 

1.2232 

1.2257 

1.2283 

1.2308 

1.2334 

1.2360 

54 

36 

1.2360 

1.2386 

1.2413 

1.2440 

1.2466 

1.2494 

1.2521 

53 

37 

1.2521 

1.2548 

1.2576 

1.2604 

1.2632 

1.2661 

1.2690 

52 

38 

1.26^0 

1.2719 

1.2748 

1.2777 

1.2807 

1.2837 

1.2867 

51 

39 

1.2867 

1.2898 

1.2928 

1.2959 

1.2990 

1.3022 

1.3054 

50 

40 

1.3054 

1.3086 

1.3118 

1.3150 

1.3183 

1.3216 

1.3250 

49 

41 

1.3250 

1.3283 

1.3317 

1.3351 

1.3386 

1.3421 

1.3456 

48 

42 

1.3456 

1.3491 

1.3527 

1.3563 

1.3599 

1.3636 

1.3673 

47 

43 

1.3673 

1.3710 

1.3748 

1.3785 

1.3824 

1.3862 

1.3901 

46 

44 

1.3901 

1.3940 

1.3980 

1.4020 

1.4060 

1.4101 

1.4142 

45 

GO' 

50' 

40' 

30' 

20' 

10' 

0' 

Deg. 

NATURAL  COSECANT. 


PKOVIDENCE,    K.    I. 


155 


NATUKAL   SECANT. 


Deg. 

0' 

10' 

20' 

30' 

40' 

50' 

60' 

45 

1.4142 

1.4183 

1.4225 

1.4267 

1.4309 

1.4352 

1.4395 

44 

46 

1.4395 

1.4439 

1.4483 

1.4527 

1.4572 

1.4617 

1.4662 

43 

47 

1.4662 

1.4708 

1.4755 

1.4801 

1.4849 

1.4896 

1.4944 

42 

48 

1.4944 

1.4993 

1.5042 

1.5091 

1.5141 

1.5191 

1.5242 

41 

49 

1.5242 

1.5293 

1.5345 

1.5397 

1.5450 

1.5503 

1.5557 

40 

50 

1.5557 

1.5611 

1.5666 

1.5721 

1.5777 

1.5833 

1.5890 

39 

51 

1.5890 

1.5947 

1.6005 

1.6063 

1.6122 

1.6182 

1.6242 

38 

52 

1.6242 

1.6303 

1.6364 

1.6426 

1.6489 

1.6552 

1.6616 

37 

53 

1.6616 

1.6680 

1.6745 

1.6811 

1.6878 

1.6945 

1.7013 

36 

54 

1.7013 

1.7081 

1.7150 

1.7220 

1.7291 

1.7362 

1.7434 

35 

55 

1.7434 

1.7507 

1.7580 

1.7655 

1.7730 

1.7806 

1.7882 

34 

56 

1.7882 

1.7960 

1.8038 

1.8118 

1.8198 

1.8278 

1.8360 

33 

57 

1.8360 

1.8443 

1.8527 

1.8611 

1.8697 

1.8783 

1.8870 

32 

58 

1.8870 

1.8959 

1.9048 

1.9138 

1.9230 

1.9322 

1.9416 

31 

59 

1.9416 

1.9510 

1.9606 

1.9702 

1.9800 

1.9899 

2.0000 

30 

60 

2.0000 

2.0101 

2.0203 

2.0307 

2.0412 

2.0519 

2.0626 

i  29 

61 

2.0626 

2.0735 

2.0845 

2.0957 

2.1070 

2.1184 

2.1300 

i  28 

62 

2.1300 

2.1417 

2.1536 

2.1656 

2.1778 

2.1901 

2.2026 

27 

63 

2.2026 

2.2153 

2.2281 

2.2411 

2.2543 

2.2676 

2.2811 

26 

64 

2.2811 

2.2948 

2.3087 

2.3228 

2.3370 

2.3515 

2.3662 

25 

65 

2.3662 

2.3810 

2.3961 

2.4114 

2.4269 

2.4426 

2.4585 

24 

66 

2.4585 

2.4747 

2.4911 

2.5078 

2.5247 

2.5418 

2.5593 

!  23 

67 

2.5593 

2.5769 

2.5949 

2.6131 

2.6316 

2.6503 

2.6694 

i  22 

68 

2.6694 

2.6883 

2.7085 

2.7285 

2.7488 

2.7694 

2.7904 

21 

69 

2.7904 

2.8117 

2.8334 

2.8554 

2.8778 

2.9006 

2.9238 

;  20 

70 

2.9238 

2.9473 

2.9713 

2.9957 

3.0205 

3.0458 

3.0715 

19 

71 

3.0715 

3.0977 

3.1243 

3.1515 

3.1791 

3.2073 

3.2360 

18 

72 

3.2360 

3.2653 

3.2951 

3.3255 

3.3564 

3.3880 

3.4203 

17 

73 

3.4203 

3.4531 

3.4867 

3.5209 

3.5558 

3.5915 

3.6279 

16 

74 

3.6279 

3.6651 

3.7031 

3.7419 

3.7816 

3  8222 

3.8637 

15 

75 

3.8637 

3.9061 

3.9495 

3.9939 

4.0393 

4.0859 

4.1335 

14 

76 

4.1335 

4.1823 

4.2323 

4.2836 

4.3362 

4.3901 

4.4454 

13 

77 

4.4454 

4.5021 

4.5604 

4.6202 

4.6816 

4.7448 

4.8097 

12 

78 

4.8097 

4.8764 

4.9451 

5  0158 

5.0886 

5.1635 

5.2408 

!  11 

79 

5.2408 

5.3204 

5.4026 

5.4874 

5.5749 

5.6653 

5.7587 

10 

80 

5.7587 

5.8553 

5.9553 

6.0588 

6.1660 

6.2771 

6.3924 

9 

81 

6.3924 

6.5120 

6.6363 

6.7654 

6.8997 

7.0396 

7.1852 

8 

82 

7.1852 

7.3371 

7.4957 

7.6612 

7.8344 

8.0156 

8.2055 

7 

83 

8.2055 

8.4046 

8.6137 

8.8336 

9.0651 

9.3091 

9.5667 

6 

84 

9.5667 

9.8391 

10.127 

10.433 

10.758 

11.104 

11.473 

5 

85 

11.473 

11.868 

12.291 

12.745 

13.234 

13.763 

14.335 

4 

86 

14.335 

14.957 

15.636 

16.380 

17.198 

18.102 

19.107 

3 

87 

19.107 

20.230 

21.493 

22.925 

24.562 

26.450 

28  653 

2 

88 

28.653 

31.257 

34.382 

38.201 

42.975 

49.114 

57.298 

1 

89 

57.298 

68.757 

85.945 

114.59 

171.88 

343.77 

00 

0 

60' 

50' 

43' 

30' 

20' 

10' 

0' 

Deg. 

NATUEAL  COSECANT. 


156 


BROWN    &   SHARPE   MFG.    CO. 


DECIMAL  EQUIVALENTS  OF  PARTS  OF  AN  INCH. 


A  - 

.01563 

«... 

.32813 

||  ...  .70313 

1 

03125 

1L  . 

34375 

II     71875 

U2  ' 

32  

A  ••• 

.04688 

If  .« 

.35938 

f|  ...  .73438 

1-16  

.0625 

o  8 

.375 

q-4.         75 

** 

A  - 

.07813 

II  - 

.39063 

|f  ...  .76563 

A  • 

.09375 

43 

.40625 

M.  .78125 

32 

2 

A- 

.10938 

ft  ... 

.42188 

U  ...  .79688 

1-8  

.125 

7-16 

4375 

13  16      8125 

A- 

.14063 

H— 

.45313 

|f  ...  .82813 

.15625 

1  5 

46875 

|7  84375 

32  •*•' 

H  — 

.17188 

ft  - 

.48438 

|{  ...  .85938 

3-16  . 

.1875 

1-2 

5 

7-8  875 

H- 

.20313 

H  - 

.51563 

IJ  ...  .89063 

_7 

21875 

(-q-l  9K 

M      90625 

7JT  



.Ool^J 

H  .~ 

.23438 

if  - 

.54688 

If  ...  .92188 

1-4.  , 

.25 

9-  16 

5625 

15-16  9375 

*1J  

ft  - 

.26563 

ft  - 

.57813 

|J  ...  .95313 

j>  

.28125 

59375 

31  96875 



ft- 

.29688 

if  - 

.60938 

||  ...  .98438 

5-16  

.3125 

8 

625 

1  1.00000 

o 

It  ... 

.64063 

IJ  

.65625 

II  — 

.67188 

11-16  

.6875 

BROWN    &   SHARPE   MFG.    CO. 

TABLE  OF  DECIMAL  EQUIVALENTS 


157 


MILLIMETRES  AND    FRACTIONS  OF  MILLIMETRES. 


mm.  Inches. 

mm.  Inches. 

mm.  Inches. 

mm.  Inches. 

jjo  =  .00039 

H  =  .  01399 

^  =  .03530 

S  =  .03740  ^ 

ifo  =  .00079 

^  =  .01339 

jjo  =  .03559 

ifo  =  .03780 

ife  =  -00118 

Wo  =  .01378 

^  =  .03598 

^f0  =  .03819 

ilo  =  -00157 

•m  =  -01417 

fj  =  .03638 

jfo  =  .03858 

llo  =  Mm 

S  —  -01457 

^  •=  .03677 

•S  =  .03898 

jfo-  =  .00336 

M  =  -OW96 

m  =  .02717 

1  =  .03937 

j^  =  .00376 

i|  =  .01535 

j^  -=  .03756 

3  =  .07874 

ifo  =  .00315 

:S  =  -01575 

^  =  .03795 

3  =  .11811 

4  =.00354 

IFo  =  .°1614 

S  =  .02835 

4  =  .15748 

^=.00394 

fo  =  -OW5* 

^0  =  .03874 

5  =  .19685 

ij=.  00433 

f0  =  .01693 

^  =  .03913 

6  =  .33633 

i>  =  .00473 

^  =  .01733 

$  -  .03953 

7  =  .37559 

Ok  =  .00513 

1)  =  -01773 

^0  =  .03993 

8  =  .31496 

^  =  .00551 

i)  =  -01811 

i,  =  .03033 

9  =  .35433 

ij  =  .00591 

^0  =  -01350 

C!  f0  =  .03071 

10  =  .39370 

^  =  .00630 

•^  =  .01890 

^  =  .03110 

11  =  .43307  L 

^  =  .00669 

100  ==s  .01920 

DA 

^  =  .03150 

13  =  .47344 

^j  =  .00709 

jg  =  .01969 

•f0  =  .03189 

13  =  .51181 

M  =  -00748 

'i)=.  03008 

S  =  -03223 

14  =  .55118 

^  =  .00787 

M  =  -02047 

83         ftQO£O 

J^Q  =  .UoJoS 

15  =  .59055 

ij  -  .00837 

Wo  =  -02087 

^0  =  .03307 

16  =  .63993 

^  ==  .00866 

S  =  -02126 

i,  -  .03346 

17  =  .66939 

H  =  .00906 

M  =  -02165 

jf0  =-  .03386 

18  =  .70866 

^  =  .00945 

^  =  .03305 

fo  =  -03435 

19  =  .74803 

1  =  .00984 

fo  =  -°2S4±  : 

jf0  =  .03465 

30  =  .78740 

^  =  .01034 

^=.03383 

Hi  =  .03504 

31  =  .83677 

^  =  .01063 

fo  -  -02323 

fo  =  .03543 

33  =  .86614 

1,  -  -01103 

CO      noo/.o 

loo  =  -0^3G3 

^0  =  .03583 

33  =  .90551 

^  =  .01143 

^  =  .03403 

-jg  =  .03633 

34  =  .94488 

S  =  -01181 

S  =  .03441 

^  -  .03661 

35  =  .98435 

Uo  =  -01220 

•^  -=  .03480 

-m  =  -03701 

36  =1.03363 

'  §t  =  .01360 

•> 

10  mm.  =  1  Centimeter  =  0.3937  inches. 
10  cm.  =  1  Decimeter  =  3.937  inches. 


10  dm.  =  1  Meter  =  39.37  inches. 
25.4  mm.  =  1  English  Inch. 


INDEX. 


A. 

PAGK 

Abbreviations  of  Parts  of  Teeth  and  Gears 4 

Addendum 2 

Angle,  How  to  Lay  Off  an 88,  105 

Angle  Increment 104 

Angle  of  Edge 100 

Angle  of  Face 102 

Angle  of  Pressure 135 

Angle  of  Spiral. Ill 

Angular  Velocity 2 

Annular  Gears 32,  137 

Arc  of  Action 136 

B. 

Base  Circle 11 

Baseof  Epicycloidal  System 25 

Base  of  Internal  Gears 137 

Bevel  Gear  Blanks 34 

Bevel  Gear  Cutting  on  B.  &  S.  Automatic  Gear  Cutter 52 

Bevel  Gear  Angles  by  Diagram 36 

Bevel  Gear  Angles  by  Calculation 100,  104 

Bevel  Gear,  Form  of  Teeth  of ....  41 

Bevel  Gear,  Whole  Diameter  of 36,  102 

C. 

Centers,  Line  of.. 3 

Chordal  Thickness 142 

Circular  Pitch,  Linear  or 4 

Classification  of  Gearing 5 

Clearance  at  Bottom  of  Space 6 

Clearance  in  Pattern  Gears g 

Condition  of  Constant  Velocity  Eatio 2 

Contact,  Arc  of 136 

Continued  Fractions 130 

Coppering  Solution 85 

Cutters,  Ho\v  to  Order 83 

Cutters,  Table  of  Epicycloidal 84 


160  INDEX. 

PAGE. 

Cutters,  Table  of  Involute 82 

Cutters,  Table  of  Speeds  for 81 

Cutting  Bevel  Gears  on  B.  &  S.  Automatic  Gear  Cutter 52 

Cutting  Spiral  Gears  in  a  Universal  Milling  Machine 120 

D. 

Decimal  Equivalents,  Tables  of ,  15G 

Diameter  Increment 102 

Diameter  of  Pitch  Circle 6 

Diameter  Pitch 5 

Diametral  Pitch 17 

Distance  between  Centers 8 

E. 

Elements  of  Gear  Teeth 5 

Epicycloidal  Gears,  with  more  and  less  than  15  Teeth 30 

Epicycloidal  Gears,  with  15  Teeth 25 

Epicycloidal  Eack 27 

F. 

Face,  Width  of  Spur  Gear 80 

Flanks  of  Teeth  in  Low-numbered  Pinions 20 

G. 

Gear  Cutters,  How  to  Order 83 

Gear  Patterns 8 

Gearin g  Classified 5 

Gears,  Bevel 34,  41,  100 

Gears,  Epicycloidal 25 

Gears,  Involute 9 

Gears,  Spiral 107,  120 

Gears,  Worm 63 

II. 
Herring-bone  Gears 128 

L 

Increment,  Angle 104 

Increment,  Diameter 102 

Interchangeable  G  ears 24 

Internal  or  Annular  Gears 32,  137 

Involute  Gears,  30  Teeth  and  over 9 

Involute  Gears,  with  Less  than  30  Teeth 20 

Involute  Rack.. 12 


INDEX.  161 

L 

PAGE. 

Lead  of  a  Worm,  .........  ........  ........................................     02 

,  Limiting  Numbers  of  Teeth  in  Internal  Gears  ..................     32 

Line  of  Centers....  ........  ...  ...........................................       2 

Line  of  Pressure./,  .......  ...  ...........................................  12,  135 

Linear  or  Circular  Pitch  ....................................  .  .......  ...       4 

Linear  Velocity.  .............  .............................................       1 

M. 

Machine,  B.  &  S.,  for  Cutting  Bevel  Gear?  ......................     52 

Module  .......................  ...  .....  ..........................................       ft 

K. 

Normal  .......  ...........  .........  .................  .  .......................   114 

Normal  Helix  .....................  ........................................    114 

Normal  Pitch  ....................  ...  ..........................  .  ............    114 

'  0. 

Original  Cylinders  .........  .  ...........................................       1 

P. 

Pattern  Gears  ..............  .  .............................................  8 

Pitch  Circle  ...........  .  ..............................  .  ...................  3' 

Pitch,  Circular  or  Linear  ........................  ,  ......................  4 

Pitch,  a  Diameter  ........  ....  ...................  ...............  .  .........  G 

Pitch,  Diametral  ..........  ................................  .............  17 

Pitch,  Normal  .....  .........  .........  ....................................  114 

Pitch  of  Spirals  ..........................................................  HO 

Polygons,  Calculations  for  Diameters  of  .........  ..................  95 

R 

Rack  ..................................  .....................  ......;...  ......  ..     12 

Rack  for  Epicycloidal  Gears  .........................  ......  .........  '.'.     27 

Rack  for  Involute  Gears  .........  .......  .'.  ..............................     12 

Rack,  for  SpiralGears  .................................  ...................  119 

Relative  Angular  Velocity  ..............................  ......  .........       2 

Rolling  Contact  of  Pitch  Circle..  .....  ........  .............  ............       3        -\ 


Screw  Gearing1.  .I....!  ...............................................  107,  128 

Single-Curve  Teeth  .........  ..........................................  ..       9 

Speed  of  Gear  Cutters  ............................................  .....  ..     81 


162  INDEX. 

PAGE. 

Spiral  Gearing 107,  120 

Standard  Templets 27 

Strength  of  Gears 140 

T. 

Table  of  Decimal  Equivalents 143,  154 

Table  of  Sines,  etc 150,  155 

Table  of  Speeds  for  Gear  Cutters 81 

Table  of  Tooth  Parts 146,  149 

V. 

Velocity,  Angular 2 

Velocity,  Linear 1 

Velocity,  Relative 2 

W. 

Wear  of  Teeth , 80,  127 

Worm  Gears 63 


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